Topology Optimisation of Composites with Base Materials of ... · Topology Optimisation of...

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Topology Optimisation of Composites with Base Materials of Distinct Poisson’s Ratios

A thesissubmitted in fulfilment of the requirements for the degree of MasterofEngineering

XuranDu

Bachelor of Engineering, China University of Mining and Technology (Beijing), China

School of Engineering.

College of Science, Engineering and Health.

RMIT University

July2017


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Declaration

I certify that except where due acknowledgement has been made, the work is that of the

author alone; the work has not been submitted previously, in whole or in part, to qualify

for any other academic award; the content of the thesis is the result of work which has

been carried out since the official commencement date of the approved research

program; any editorial work, paid or unpaid, carried out by a third party is

acknowledged; and, ethics procedures and guidelines have been followed.

Xuran Du

31 July 2017


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Acknowledgments

I would like to express my deepest appreciation to all those who provided me with the

possibility to complete this thesis. My special and sincere gratitude goes to my

supervisor Professor Mike Xie for his valuable suggestions, comments and contribution

throughout this entire research process. I would also like to express my warm

appreciation to my second supervisor, Dr. Xiaodong Huang, for his encouragement,

beneficial comments, contribution and helpful advice.

I also very much appreciate the help and guidance received throughout my candidature,

from Dr. Annie Yang and Dr. Joe Zhihao Zuo. I would also like to warmly thank the

former SCECE School Manager Ms. Marlene Mannays and Research Coordinator Mr.

Michael Jacobi for their help.

This thesis is dedicated to my parents whom I would like to deeply thank, for their

endless kindness, love and support.


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Publications list

1- Du, X., Min, Y., Yang, X., & Zuo, Z. (2014). Topology Optimisation of Composites

Containing Base Materials of Distinct Poisson ’s Ratios. Mechanics and

Materials, 553, 813–817. doi:10.4028/www.scientific.net/AMM.553.813Applied

2- Long, K., Du, X., Xu, S., & Xie, Y. M. (2016). Maximizing the effective Young’s

modulus of a composite material by exploiting the Poisson effect. Composite

Structures, 153. doi:10.1016/j.compstruct.2016.06.061


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ContentsNotations ………………………..………………….…………………………………………………………………………VIAbstract …………………………..…………………………………………………………………………………………..1Chapter1Introduction ………………………………………….....…………………………………………..….4

1.1Problemstatementandmethodology …………………………………………………………………….71.2Significance ………………………………………………………….……………………………………………..111.3Outlineofthesis …………………………………………………………………………………………….12

Chapter2Literaturereview …………………………………………………………………………………………….152.1Background ……………………………………………….………………………………………………………..172.2SIMPmethod …………………………………………..……..…………………………………………212.3EvolutionaryStructuralOptimization(ESO) ……………………………………………..……….262.4Thelevelsetmethod ……………………………………………………….……………………………………302.5Bi-DirectionalEvolutionaryStructuralOptimization(BESO) …..…………………………33

2.5.1Hard-killofelementsinBESO ………………………………………………………36 2.5.2Soft-killofelementsinBESO …………………….…..……………………………40Chapter-3-TopologyoptimisationofCompositesforMaximisingEffectiveYoung'sModuli.43 3.1Methodology………………..……………………….………………………………………………………………45

3.1.1Optimizationproblemstatement ………………………………………………………453.1.2Optimizationwithoptimumvolumetobesolved…………….…………………………463.1.3Verificationonsensitivityanalysis ………………………………………………………483.1.4Comparematerialinterpolationschemes ………………………..…….…………513.1.5CompareInitialdesigns ……………………………………………..……………………55

3.2 CasestudywithnegativePoisson’sratio ………………………………………………………583.3Concludingremarks …………………….………………………………………………………………………61

Chapter-4-TopologyoptimisationofcompositesforMaximisingBulkorShearModulus….62 4.1Methodology ……………………..………..…….…………………………………………….………64 4.1.1Problemstatementofperiodicmaterialtopologyoptimization……………………64 4.1.2TopologyoptimizationthroughBESOmethod ………………………………….…..…66 4.1.3Homogenizationandsensitivityanalysis ………………………………….…..…68 4.1.4Numericalinstabilitiesandfilteringscheme ……………………..………….………784.1.5BESOProcedure …………………………………………………………………….…………80 4.2Resultsanddiscussion …………………………………………..…………………………..………83 4.2.1Compositeswithmaximumbulkmodulus ……………………….…………………83 4.2.2Compositeswithmaximumshearmodulus …….………………………..…………89 4.2.3ValidationusingHSWtheoreticalbound …………….……………………..……93

4.3Concludingremarks ………………………………………………………………………………….………..99Chapter-5-Conclusionsandrecommendations………………………………….…………………….………101 5.1Conclusions ……………………………………………………..……………………………………….………101 5.2Recommendations ……………………………………..………………………..……………………..…103References ………………………………………………………………………………….………………….……………104


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Notations:

iα̂ Filtered sensitivity number of element i

iα~ Average sensitivity number with values of previous iteration i

iα Sensitivity number of elements i ε

Strain tensor ν Poisson’s ratio σ Stress tensor

ijσ

Stress tensor Ω Design domain B Strain-displacement matrix c Constant number d Dimension of the model D Stiffness matrix ijklE Elasticity tensor sijklE Elasticity tensor of the base material HE

Material’s homogenized elasticity tensor ER Evolution rate Ei Young’s modulus of base material i

G Material shear modulus defined as average shear modulus along principal axes

Gi Shear modulus of base material i I Unit matrix K

Bulk modulus

Ki

Bulk modulus of base material i N Number of finite elements in structural model

p Penalty exponent r ij Distance between element i and element j r min

Filter radius

t Iteration number

iu Displacement vector V Volume *V Prescribed volume iV Volume of element i

fV Volume fraction of a certain phase in composite materials

ix

Design variables of element i

minx Lower bound of design variable

ijx Design variable which indicates the density of the thi element for the thj material

Note: Specific notations are defined by various subscripts or superscripts; see definitions in the text.


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Abstract:

Structural topology optimization approaches help engineers to find the best layout or

configuration of members in structural systems. However, these approaches differ in

terms of computational costs and efficiency, quality of generated topologies, robustness,

and the level of effort for implantation as a computational post-processing procedure.

On the other hand, one common approach for saving resources is the application of

porous or composite materials that have extreme or tailored properties. It is known that

composite materials with improved properties can be designed by modifications into the

topology of their microstructures. A systematic way for improving the properties of

these types of materials consists of application of a structural topology optimization

approach to find the best spatial distribution of materials within the microstructures of

composites.

This study presents new approaches for design of microstructures for materials based on

the bidirectional evolutionary structural optimization (BESO) methodology. It is

assumed that the materials are composed of repeating microstructures known as

periodic base cells (PBC). The goal is to apply the BESO topology optimization to find

the best spatial distribution of constituent phases within the PBC in such a way that

materials with desired or improved functional properties are achieved. To this end, the

homogenization theory is applied to establish a relationship between material properties

in microstructural and macrostructural length scales.


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In the first stage of this study, the optimization problem is formulated to find

microstructures for composites of prescribed volume constrains with maximum

effective Young's moduli. It is assumed that the base materials are composed of two

materials with different Poisson’s ratios. By performing finite element analysis on the

PBC and applying the homogenization theory, an elemental sensitivity analysis is

conducted. Following by removing and adding elements gradually in an iterative

process according to their sensitivity ranking, the optimal topology for the PBC can be

generated. The effectiveness and computational efficiency of the proposed approach is

numerically exemplified through a range of 3D topology optimization problems.

In the next stage of this study, the optimization problem is formulated to find

microstructures for composites of prescribed volume constrains with maximum stiffness

in the form of bulk or shear modulus. Here the composites possess two constituent

phases. Compared with cellular materials whose microstructures are made of a solid

phase and a void phase, composites of two different material phases are more

advantageous since they can provide a wider range of performance characteristics.

Maximization of bulk or shear modulus subject to a volumetric constraint is selected as

the objective of the material design. Adding and removing of elements is performed

based on the ranking of sensitivity numbers and imposed volumetric constraint between

different base materials. The proposed procedure demonstrates very stable convergence

without any numerical difficulty. The computational efficiency of the proposed

approach has been demonstrated by numerical examples. A series of new and

interesting microstructures of two base materials are presented. The other major


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advantage of the BESO in design of composites of two base materials is the distinctive

interfaces between constituent phases in the generated microstructures, which make the

manufacturing of microstructures viable. The methodology has the capability to be

extended for material optimization with other objective or constraint functions.


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Chapter1Introduction

The main objective of structural engineering is to develop load-carrying systems that

can economically satisfy the design performance objectives and safety constraints.

Economical consideration is the main motivation for the developing of design process

that enables the minimization of the resource consumption. In fact, many engineering


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disciplines are involved in optimization and use the mathematical language for this

purpose. For optimization of structures, this goal can be achieved by finding the best

topology, layout of members or material distribution within the design domain of the

structural system.

The history of the structural optimization can be traced back to Michell’s (Michell

1904) theoretical studies on optimality conditions of structural systems in Melbourne,

Australia. However, the early studies were mainly remained limited to the size and

shape optimization of predetermined topologies. Wider access to computational

machines in 1990s justified the development of numerical procedures for the topology

optimization of structures which aims at finding the best layout, configuration and

spatial distribution of materials in the domain of the continuum structure (Bendsøe and

Kikuchi 1988; Rao 1995; Burns 2002; Schramm and Zhou 2006). It was not so long

afterwards when the first topology optimization commercial software packages such as

“Altair OptiStruct” emerged (Schramm and Zhou 2006). Since then refining the

theories and developing of new methods are among active fields in structural

engineering.

In addition to topology optimization of structures in macro-scale, one common

approach for saving resources is the application of porous or composite materials that

have extreme or tailored properties. In fact, the responses of structural systems are

highly dependent on the material they are built from. Although application of composite

materials in structures had a rapid development in the past few decades, the idea of

combining materials in order to achieve improved characteristics is not new. In fact,

material science is one of the oldest forms of applied science. For example, smelting


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and casting metals can be traced back to the Bronze Age. Egyptians smelted iron for the

first time at approximately 3500BC, which possibly happened as a by-product of copper

refining. Iron was used in tiny amounts mostly for ornamental or ceremonial purposes at

that time. However that marked the first milestone of what will become the world's

dominant metallurgical material.

There are different reasons for the demands for materials with tailored or improved

properties, and a variety of performance demands in terms of functional properties are

being placed on material systems. These include lightweight materials with improved or

tailored mechanical, thermal, optical, flow, chemical, and electromagnetic properties

(Evan 2001; Torquato 2002). For instance, applications of lightweight multifunctional

products in vehicles save energy in terms of lower fuel costs and can reduce the gas

emission damages to the environment significantly.

Traditionally, the objectives of material design are achieved by application of

composites in the form of fibre, particulate or laminar (Figure 1.1), in which the

properties of materials is controlled by modifying the location, material constituents,

orientation, or volume fraction of fibre, particles or laminar inclusions (Staab 1999).

The traditional material design method follows the fundamental trial and error method

where design changes are made and the material is re-analysed repeatedly until its

performance meets the desirable objectives (Torquato 2010). Although material design

has achieved its objectives in certain cases through this approach, the desire for

development of systematic approaches has made the material design an active field of

research (Cadman, Zhou et al. 2012).


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Figure 1.1 Composite classes

(a) Composite (b) Particulate Composite (c) Laminar Composite (d) Cellular Composite

Materials with repeating or periodic microstructures are usually consisted of one

constituent phase and one void phase, also known as porous or cellular materials, or

combinations of two or more different constituent phases with or without void phase,

also known as periodic composites (Huang, Xie et al. 2012). The overall properties of

these types of materials are controlled by the spatial distribution of constituent phases

within the PBC, as well as properties of selected constituent phases. In comparison with

traditional composites, periodic composites demonstrate greater flexibility in terms of

capability to be tailored for prescribed physical properties by controlling the

compositions and microstructural topology of the constituent phases (Cadman, Zhou et

al. 2012). They can also be easily tailored to have gradation in functional properties in

the form of an FGM, through gradual changes in their periodic microstructural

topologies( a. Radman, Huang, & Xie, 2012).

1.1. Problem statement and methodology

The periodic base cell (PBC) could be viewed as a heterogeneous continuum structure

that is composed of domains of different constituents phases, where a phase is a single

type of material (Bendsøe and Sigmund 2003). It is shown that the properties of

materials are influenced by the topology of the PBC (Hassani and Hinton 1998; Hassani

and Hinton 1998; Hassani and Hinton 1998). Hence, a major challenge in design of


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these types of materials would be the determination of the optimum spatial distribution

of constituent phases within the microstructure. In the simplest form, the periodic

composite materials consist of one base material as “shell” and the other one as

inclusion, also known as in fractal foam form. Therefore, the PBC could be viewed as a

structure and it is reasonable to apply the structural topology optimization

methodologies for determination of the spatial distribution of the phases.

Along with the development of computer technology, progress in the area of numerical

methods is often ahead of mathematical approaches. The reason is largely attribute to

the fact that the mathematical approaches usually require exhaustive formulation and

rigorous solution to the corresponding rather simple optimization problems while in

numerical approaches complicated models could be dealt with rather simple principals

(Cherkaev 2000). On the other hand, numerical topology optimization usually engages

with large numbers of design variables that makes the conventional mathematical

optimization algorithms inappropriate, as they may not be efficient enough to solve the

problems with large heterogeneity, mainly as the result of high time consumption

(Cadman, Zhou et al. 2013). In the past two decades, several numerical topology

optimization algorithms have been examined with the goal of developing a systematic

approach for design of periodic materials. One of the main concerns in these attempts

was the computational efficiency of the approach.

Basically, the topology optimization techniques, such as homogenization method

(Bendsøe and Kikuchi 1988), level set method (Wang, Wang et al. 2003; Wang, Wang


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et al. 2004), solid isotropic material with penalization (SIMP) (Bendsøe 1989; Zhou and

Rozvany 1991; Rozvany, Zhou et al. 1992), evolutionary structural optimization (ESO)

(Xie and Steven 1993; Xie and Steven 1997), and bi-directional evolutionary structural

optimization (BESO) (Querin, G.P.Steven et al. 1998; Yang, Xie et al. 1999; Huang

and Xie 2007a; Huang and Xie 2010a) were developed to find the stiffest structural

layout under the given constraints. Prior to the commencement of this research, SIMP

(Sigmund 1994; Sigmund 1995), level set (Wilkins, Challis et al. 2007; Challis, Roberts

et al. 2008; Zhou, Li et al. 2010), ESO (Patil, Zhou et al. 2008) and BESO (Huang,

Radman, & Xie, 2011; Huang & Xie, 2007, 2008; A. Radman, 2013) have been

extended into the design of periodic microstructures of materials.

Different topology optimization techniques offer advantages and disadvantages in terms

of computational costs and efficiency, quality of generated microstructures, robustness,

and the level of effort for implantation as a computational post-processing procedure, to

name a few. Among various topology optimization algorithms, ESO (Xie and Steven,

1993, 1997) was originally developed based on the concept of gradually removing

inefficient elements from the finite element model of the structure so that the resulting

topology evolves towards an optimum. A later version of the ESO method, namely the

bi-directional evolutionary structural optimization (BESO) (Querin et al. 1998; Yang et

al. 1999) allows removing elements from the least efficient regions, and adding

elements to the most efficient regions of the finite element model of the structure.

Further developments on BESO have been made by theoretically introducing the hard-

kill BESO (Huang and Xie, 2007) and soft-kill BESO (Huang and Xie 2009, 2010a)


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under certain circumstances. The new soft-kill BESO (Huang and Xie 2007a) improved

most of the deficiencies of previous versions (Rozvany, 2009; Huang and Xie 2010b). It

offers several advantages in comparison with other topology optimization algorithms in

terms of quality of the generated topology and convergence speed.

This study is the first attempt to extend the application of the BESO into the design of

microstructures of materials with different Poisson’s ratio. Since materials with high

stiffness are more desirable from structural application point of view, the first step of

this study is the development of the new algorithms for designing composite materials

with extreme Young’s moduli. Thereafter, the methodology will be extended into other

scenarios of material design, like pursuing extreme bulk or shear modulus. For this

purpose new procedures will be purposed for design of two-phase composite materials

with base materials possessing different Poisson’s ratio.

In particular the objectives of this study are:

• Development of computational algorithm for topological design of two-phase

composite materials with base materials possessing different Poisson’s ratio with

extreme Young’s moduli;

• Development of computational algorithm for topological design of two-phase

composite materials with base materials possessing different Poisson’s ratio with

extreme bulk or shear modulus;

It should be stated that the properties of materials varies by their chemical and atomic

configurations as well as by their especial microstructural topology (Mercier, Zambelli


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et al. 2002). However, this study deals with the materials which their microstructural

length scale is much larger than the atomic dimensions and also considerably smaller

than the overall dimensions of the structure; therefore, it is assumed that the interatomic

forces are negligible.

1.2. Significance

As it was discussed earlier, the performance enhancement of materials will lead to

significant saving of energy and resources. For instance, lightweight materials can save

energy in terms of lower fuel and emissions usage, thus reducing our carbon

discharging and considered as a greener choice. The demand for new materials with

improved functional properties is constantly increasing. As the consequences, this

growth necessitates the development of more advanced design tools. In case of periodic

materials, as the problem involves continuum structures with large heterogeneity, this

objective could be achieved by application and development of appropriate structural

topology optimization methods.

In spite of the fact that the new BESO procedure is developed very recently, the method

has acquired great successes in solving topology optimization problems in different

areas of structural engineering such as minimizing structural volume with a

displacement or compliance constraint (Huang and Xie 2009b; Huang and Xie 2010c),

stiffness optimization of structures with multiple materials (Huang and Xie 2009a),

design of periodic structures (Huang and Xie 2008), structural frequency optimization

(Huang, Zuo et al. 2010d), optimization for energy absorbing structures (Huang, Xie et

al. 2007), solving geometrical and material nonlinearity problems (Huang and Xie


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2007; Huang and Xie 2008) and design of functionally graded cellular materials ( a.

Radman et al., 2012).

This study will extend the application of BESO into the design of two-phase composite

materials with base materials possessing different Poisson’s ratio and introduces new

methodology for solving engineering problems related to composite materials. The

outcomes signify the theoretical importance of the research. On the practical side, the

advantages of BESO in simplicity, versatility and ease of implementation will provide

engineers with a new methodology and an advanced design tool for exploration and

creation of novel materials that perform the required functions.

More importantly, the previous studies on material design through structural topology

optimization methodologies have indicated that the generated micro-structural

topologies are highly dependent on the applied optimization algorithm and parameters

(Sigmund 1994; Neves, Rodrigues et al. 2000). The reason attributes to the fact that a

number of topologically different microstructures could provide similar material

property. In other words, there is no unique solution and there might be many local

optima in design of microstructures for materials. Therefore, it is important to attempt

new and different optimization algorithms, such as BESO, in order to find a much wider

range of possible solutions to material design.

1.3. Outline of thesis

This study deals with the topology optimization of microstructures for materials

therefore in the next chapter a review on various structural topology optimization


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techniques will be presented. The process of material design involves with

determination of material properties through the modelling of its representative volume

element (RVE). Chapter two also briefly introduces the related methods. This is

followed by a brief summary of previous researches on the applications of structural

topology optimization methodologies in design of microstructures for materials in the

order of SIMP, ESO, the level set method, and BESO.

Chapter three deals with the topology optimization of materials with base materials

possessing different Poisson’s ratio with extreme Young’s moduli, using the BESO

technique. As the first step in this chapter, composite materials whose base materials

possessing Poisson’s ratio above zero are considered. The statement of the optimization

problem will be presented and the details of design algorithms will be explained. The

result of applying such procedures will be presented by numerical examples. Later in

this chapter, the application of the same algorithms will be extended to composite

materials whose base material/materials possessing Poisson’s ratio between zero and

minus one.

Chapter four examines the possibility of design two-phase composite materials with

base materials possessing different Poisson’s ratio with extreme bulk or shear modulus.

Compared with cellular materials whose microstructures are made of a solid phase and a

void phase, composites of two material phases are more advantageous since they can

provide a wider range of performance characteristics (Zhou and Li 2008b). A new

computational code was carried out in this study. After presenting the details of the


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proposed method, numerical examples will be presented and compared with literature to

support the validity of the procedure.

Finally, some conclusions from this thesis are summarised in Chapter five and some

further work are also recommended in this chapter.


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Chapter2Literaturereview

There are two aspects to regulate material properties. From the chemical engineering

point of view, change their compositions; and from the civil engineering point of view,

alter their microstructural topology. Here in this study we work from the microstructural

topology optimization aspect. Some milestone works in microstructural topology

optimization research are listed below as a general review. And more details are

contained in the following sections.


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Bendsøe et al. ((Bendsøe et al. 1993) proposed an analytical model aimed at predicting

optimal material properties, which demonstrated the possibility of designing materials

topology such that materials containing extreme properties can be achieved.

Following this work, a computational algorithm was formulated by Sigmund (Sigmund

1994a; Sigmund 1994b; Sigmund 1995) to solve the problem of determining the

microstructure for such material with given homogenized properties. The Sigmund

algorithm is based on a structural topology optimization technique and the methodology

has been referred to as “inverse homogenization” (Sigmund 1994; Steven 2006;

Cadman, Zhou et al. 2012).

Following the development of the Sigmund algorithm, several structural topology

optimization methods have been researched and developed to design the microstructural

topology for materials. Solid isotropic material with penalization method (SIMP) was

the method used in the work by Sigmund (Sigmund 1994a; Sigmund 1994b; Sigmund

1995). In the same decade, another structural topology optimization method, the

evolutionary structural optimization (ESO) method (Xie and Steven 1993) was

introduced to the world. Shortly after, other structural topology optimization methods

include the level-set method (Sethian and Wiegmann 2000; Osher and Santosa 2001;

Wang et al. 2003) and bidirectional evolutionary structural optimization (BESO)

method (Huang and Xie 2007) have also been applied for topology optimization of

structures/materials. Quality of the microstructures generated and efficiency are major

concerns when comparing different topology optimization theories. While


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computational costs, robustness, and the applicability as a post-processing procedure are

some of the key differences between their computational algorithms and applications.

This thesis is dedicated to investigate designing composite materials with periodic

microstructures possessing base materials of different Poison’s ratio in order to

maximize its Young’s/bulk/shear modulus by a topology optimization approach. The

method used is based on bidirectional evolutionary structural optimization (BESO)

(Huang and Xie 2007) technique. The optimization problem is formulated as finding a

microstructural topology with the maximum Young’s/bulk/shear modulus under a

prescribed volume constraint and it is solved by a searching algorithm based on

sensitivity analysis. The effect of interpolation function in the sensitivity analysis is

studied then applied numerically within a periodic base cell (PBC) by gradually

removing and adding elements. Examples of different combinations of base materials

demonstrate the effectiveness of the proposed method for achieving convergent

composite periodic materials with optimal Young’s/bulk/shear modulus. Results show

some interesting topological patterns that can be used for guiding periodic composite

material design.

This chapter offers a critical review of the structural topology optimization approaches

that have so far been researched and applied to material design field.

2.1. Background

Human beings stood out from animal kingdom with advantages including a relatively

larger brain that enabled high levels of abstract reasoning and problem solving. Human


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beings, regardless of the era or location they lived in / are living in, their thoughts share

an everlasting subject, which is seeking the optimums. The pursuit of optimal solutions

is to boost productivity, fully acknowledge and implement the values of all matters, and

satisfy aesthetic tastes, as well as an instinct that originates from the human spirit in the

pursuit of perfection. It is embodied in every human action, from how we carry out

small tasks on daily basis to the field of scientific research.

In scientific research, the action of seeking the optimal solution, a.k.a. optimization,

started with mathematical analysis that was gradually formed and developed with the

establishment of civilization, and joined by numerical analysis in the last century.

Calculus of variations uses the solution of differential equations to identify optimal

points in a function, where the maximum and minimum value of said function is

represented as differential equations. Under the circumstance of very simple cases, the

calculus of variations provides an effective method of solving for problems involving

extremization. Although this is not true for non-linear differential equations, in which

case the chance of obtaining a closed form solution is very unpredictable. In contrast,

numerical approaches for solving variational equations are partly based on

approximation of derivatives, which suffer from problems in structural optimization in

the form of time consumption, accuracy, and convergence of outcomes (Kamat 1993).

(Michell 1904) was the first to introduce the theory of structural optimization for the

development of minimum weight truss-like structures in Australia (Eschenauer &

Olhoff 2001), but it was only till later during the 1950s when the idea of structural


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optimization became widespread with the development of digital technology especially

computers. This boosted the development of linear programming methods (Dantzig

1963), which successfully solved a range of structural optimization problems and

therefore provided significant improvements to the theory (Prager 1969; Prager 1974;

Save 1975).

Topology optimization, which is also referred to as layout optimization or generalized

shape optimization (Olhoff and Taylor 1979; Rozvany, Zhou et al. 1992; Haber,

Bendsøe et al. 1996; Eschenauer and Olhoff 2001), is intended for determining the

optimum topology, layout or configuration in the domain of a continuum structure

(a.k.a. Design Domain). Mathematically speaking, all subsets of the three dimensional

space (lines, curves etc.) can be seen as topological domains. Likewise in the field of

structural engineering, topological domains and topology basically describe the spatial

distribution of materials or location of members and joints in a structure.

With the introduction of the ground structure in which mathematical programming (MP)

algorithms were used, topology optimization was further improved during the 1960s

(Dorn, Gomory et al. 1964; Tanskanen 2002). Subsequently, the “optimal layout

theory” introduced by Prager (Rozvany 2009) and stiffness maximization of solid plates

with volumetric constraints by Cheng and Olhoff (Cheng, Olhoff 1981) are some of the

other remarkable earlier works on topology optimization. Bendsøe and Kikuchi

(Bendsøe, Kikuchi 1988) later introduced the “homogenization method” derived from a

finite element method as the first numerical structural topology optimization technique.


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This field of study was further developed by Xie and Steven (Xie & Steven, 1993) by

the “evolutionary structural optimization” method, another method based on the finite

element topology optimization.

The minimization and maximization of a defined performance function subject to a set

of constraint conditions are often encountered in structural topology optimization

problems (Kamat 1993). In general, the variables are defined as either the quantities that

define the geometry of the physical system and/or the sizes of the structural elements.

To illustrate, the topology optimization of a continuum may consist of the determination

for every point in space, existence, or absence of material in such a way that the

objective function is extremized and the constraints are satisfied, if each point in the

domain of a continuum structure (design domain) can be considered either a material or

void (Kamat 1993).

The structural optimization methods often employ simplified mathematical equations,

solved in an iterative numerical procedure. This is contrary to the classical

mathematical optimization methods, which make use of differential equations for

solution. Commonly, the following steps are involved in a basic maximization problem:

1. Assigning initial design variables (e.g. material types, weight of a structure).

2. Evaluating the objective function for the current set of design variables.

3. Comparing current properties to the prescribed values.

4. Updating the design variables with the purpose of improving the objective

function through a certain procedure.


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5. Repeating steps 2 to 4 until no further improvement in results are obtained.

In order to update the design variables, a number of approaches can be taken, including

methods that randomly select new design variables or methods that use the derivative of

the objective function to obtain the optimum. Of note, the selection of initial topology

or the procedure of updating the design variables may lead to a solution which is a local

optimum. The number of repeats will still be affected even if the solution has one global

optimum with no local optima, using this procedure.

In this study, BESO method is applied. In the following parts, BESO will be compared

with other optimization techniques that have previously been used in the design of

microstructures of materials according to the timeline of their introduction – SIMP,

ESO, and Level-set method. Since the ESO is technically early works of BESO, the

attention here is more focused on these two methods as theoretical foundation of chapter

three and four.

2.2 SIMP method

Rossow and Taylor (1973) initiated the idea of finite element based material distribution

method in topology optimization in their studies through the use of continuous design

variables without penalization of intermediate densities (Rozvany 2009). The principals

of SIMP for topology optimization of structures were first proposed by Bendsøe (1989)

in a separate study inspired by the homogenization method. Bendsøe named this method

“the direct approach” (Bendsøe 1989). Rozvany et al. (Rozvany et al. 1992) later

thought up the name SIMP, short for ‘Solid Isotropic Microstructures with Penalization',


22

which was picked up by Bendsøe and Sigmund, where `M' stood for `Material'

(Bendsøe and Sigmund 1999). This method was vastly used for the design of

microstructures for materials (Sigmund 1994a; Sigmund 1994b; Sigmund 1995).

The objective of topology optimization in continuum structures via material distribution

is to a solid or void property to each point of space (Bendsøe 1989). These problems are

generally handled by discretizing the continuum structures into a finite element model,

allowing for the change of the topology without the need of meshing in between any

two iterations. To put it mathematically, here we use the simplest form: only a single

objective function f(x) and no other performance constraints. The structural topology

optimization problem can be expressed as:

Minimize: f(x)

(2.1) Subject to:

1or 0=ix

Where V* is the prescribed volume of structure. Vi is the volume of element i. The

design variable xi denotes whether an element is present or absent, as 1 to be present and

0 to be absent. A similar formulation was suggested by Kohn and Strang (1986) for the

point-wise material/no material (also known as black/white (Bendsøe and Sigmund

1999)) optimization. However, this type of problems is ill-posed and would be

dependent on the selection of the sizes of elements and the discretization mesh,

confirmed by examination carried out by Bendsøe and Kikuchi (Bendsøe and Kikuchi

01

* =−∑=

N

iii xVV


23

1988). In one example it was shown that given higher mesh density in the finite element

model of designed structure, the optimization procedure resulted in designs containing

more members of smaller sizes therefore convergence was not achievable by using even

finer mesh sizes (Bendsøe and Sigmund 1999; Huang and Xie 2010a).

The SIMP method overcomes these issues by using a relaxation method where the

design variables are freed to take any value between 0 and 1 (Sigmund and Petersson

1998) whereby some form of penalization approach then directs the solution to a

discrete value of 0 or 1. This new definition of the optimization problem can be

expressed as following:

Minimize: f(x)

(2.2)Subject to:

10 ≤≤< imin xx

where lower bound xmin is defined by density so as to avoid singularity of the

equilibrium equations. Sigmund and Petersson applied this definition to some energy

form of the structural properties (e.g. compliance). Bring back into the above equation

(2.2), it is clear that in the new formulation of the problem, the energy property was

linearly dependent on the design variable (Sigmund and Petersson 1998).

The main concern with SIMP topology optimization is in defining relationships between

materials properties (such as Young’s modulus) and the continuous design variables. To

build the link with certain material property, first we should match the design variable

01

* =−∑=

N

iii xVV


24

with a physical property, which procedure is also known as the interpolation scheme.

As stated previously, the design variable is often interpreted as elemental density.

Bendsøe (Bendsøe 1989) used the power law approach as material interpolation scheme

in his original study in 1989. The material interpolation scheme there was put in a

simple form, where the local material elasticity was interpolated as:

Eijkl (xi ) = xipEijkl

s (2.3)

where Eijkls was the elasticity tensor ijklE of the base material. When the penalization

factor was chosen as p=1, the intermediate values of design variables (gray elements)

would possibly exist in the model. While when the penalization factor was increased to

p >1, the values of design variables were suppressed, so the stiffness tended to be very

close to 1 or 0. It is notable that a pure 0/1 solution is still impossible and some grey

elements will always remain in the structure.

Numerical experience indicates that for cases in which the volume constraint is active,

the solution comes very close to a 0/1 design given that a sufficiently large p is selected.

This is attributed to the fact that the volume remains linearly proportional to , whereas

the intermediate densities are suppressed in stiffness calculations, and stiffness would

become less than proportional (Bendsøe and Sigmund 1999). Therefore, it is important

to select a large enough value for p.


25

On the other hand, the interpolation scheme of equation (2.3) does not guarantee that

the volume distribution relationship

∑=

=N

iii xVV

1 (2.4)

stands correctly for a real composite material with specific volume. Nevertheless, it is

possible to establish conditions on p such that the power-law scheme portrays a

meaningful physical interpretation. It was shown that the power-law model achieves a

real physical interpretation if the following equations hold true (Bendsøe and Sigmund

1999):

in 2D cases (2.5.a)

and

in 3D cases. (2.5.b)

Here is the Poisson’s ratio. Zhou and Li applied the analytical bounds on materials’

properties (e.g. Hashin-Shtrikman bounds (Hashin and Shtrikman 1963)) as the

interpolation scheme (Zhou and Li 2008f), in place of using the power law scheme with

penalty exponent. In this way the necessity of determining the penalty exponent can be

eliminated.

⎭⎬⎫

⎩⎨⎧

+−≥

νν 14,

12maxp

⎭⎬⎫

⎩⎨⎧

−−

−−≥

)21(2)1(3,

57115max

νν

ννp


26

A number of solution algorithms are available for structural optimization based on finite

elements (Coville 1968; Asaadi 1973; Schittkowski, Zillober et al. 1994; Chen, Silva et

al. 2001). Among them there are the “Method of Moving Asymptotes” and the “optimal

criteria methods” as two classes of numerical approaches commonly used along with

the SIMP method.

2.3 Evolutionary Structural Optimization (ESO)

A method based around the idea of gradually removing inefficient materials from the

finite element model was introduced by Xie and Steven (Xie and Steven 1993) and this

was termed the evolutionary structural optimization (ESO). The approach achieved

great acceptance due to its simplicity, leading to extensive study (Burns 2002) and

progressive development in the form of solving stiffness and displacement problems

(Chu, Xie et al. 1996), dynamic analysis of structures (Xie and G.P.Steven 1996; Zhao,

Steven et al. 1997), buckling analysis (Manickarajah, Xie et al. 1998) or multi-criteria

optimization (Proos, Steven et al. 2001). Bi-directional evolutionary structural

optimization (BESO), which was a result of studies on ESO by Querin, G.P.Steven et

al. (1998) is also considered an important development. Patil, Zhou et al. (2008)

recently employed ESO in the design of microstructures for materials to attain the

desired thermal conductivity.

Failure of a structure occurs in the event that the stress or strain at some elements

exceeds maximum values. On the other hand, low stress or strain elements can be

treated as ineffective materials. From these arguments, it can be said that identical


27

levels of stress should exist in every element in an ideal structure (Burns 2002). From

there, we can set the rejection criteria to be based on the stress level in elements. There

were several stress indicators taken into consideration and during the early stage of

stress-based ESO method (Xie and Steven 1993) the rejection criteria was based on von

Mises stress in elements of the structure which is an indicator of average stress in each

element. In two-dimensional (2D) problems, the von Mises stress can be expressed as:

2122211

222

211 3σσσσσσ +−+=ve (2.6)

In the early stage of ESO method, the von Mises stress of each element was compared

with the maximum von Mises stress of the structure . The von Mises stress in

elements here was determined by finite element analysis (FEA). At the end of each

FEA the elements that satisfied the following condition would be removed from the

finite element model of the structure:

tv

ve RR<maxσσ

(2.7)

Here tRR is the rejection ratio at iteration t. The iterative procedure proceeds until it

reaches a “steady state” condition where there are no longer any elements to be removed

from the structure. The stress level of all elements in the structure will be greater than

at steady state. If required, the rejection ratio is increased at this stage by the

evolutionary rate (ER) which is an initial parameter defined into the ESO:

ERRRRR tt +=+1 (2.8)


28

Following the increase, this process is repeated until a new steady state is reached. Once

the structure reaches the required stress level, the procedure is terminated; say there are

no more elements with the stress level less that 20% of the maximum stress. Still, this is

not the ideal solution and only in a limited number of cases can a fully stress structure

be achieved (Burns 2002).

Another criterion for element removal could be based on the sensitivity numbers, which

is a measurement to determine the impacts of individual elements on the changing of

the objective function. In the case of optimization for compliance, the sensitivity of

elements was applied instead of using stress level in the original ESO as the elements

removal criteria (Chu, Xie et al. 1996). It can be expressed as:

iiTii uKu=α (2.9)

Where Ki is the element stiffness matrix and ui is the displacement vector of the ith

element from the output of finite element analysis of the structure. In essence, the

optimization algorithm used in compliance-based procedure is identical to the stress-

based ESO ---- the only change being that maxα and iα are replaced with vmaxσ and vσ

respectively. Notably, no noticeable discrepancies exist between the topologies obtained

via the stressed-based ESO and the compliance based approach (Li, Steven et al. 1999).

The ESO method is not only based on intuitive methodology but also proved

mathematically feasible. An attempt to explain the validity of the approach

mathematically was conducted by Tanskanen (2002) who studied the theoretical bases


29

of the compliance-based ESO. It was concluded that the ESO actually minimizes the

product of mean compliance and volume. If the design domain is modeled using equally

sized elements, the ESO was found to be similar to the sequential linear programming

method (SLP) optimization method (Tanskanen 2002).

The numerical instabilities such as checkerboard pattern and mesh dependency in the

ESO method can be avoided by developing a smoothing algorithm by averaging the

sensitivity of elements with the sensitivities of surrounding elements (Li, Steven et al.

2001). The simplicity of ESO as a topology optimization approach in both theory and

application becomes its main advantage. The approach is easily applied as a post-

processing algorithm to most finite element packages. Furthermore, the approach is

more cost-effective since the size of the finite element model is reduced as elements are

gradually removed. Besides that, the results are easier to interpret since the produced

topology is made up of a clear distinctive region, without gray areas. On the other hand,

recovery is unfeasible in the ESO approach if some elements are accidentally removed

from the structure (Zhou and Rozvany 2001). To circumvent such situations in ESO, it

is usually necessary to use very small evolutionary rates, at the expense of more costly

optimization. This means that although ESO is capable of significantly improving the

initial topology, however in some cases the results may not certainly be a global

optimum (Huang and Xie 2010; Huang and Xie 2010).


30

2.4 The level set method

The level-set method is a mathematical concept introduced by Osher and Sethian (1988)

for computation of moving interfaces (Burger and Osher 2005). Recently, it has been

used as a numerical alternate procedure to material distribution methods for structural

topology optimization (Sethian and Wiegmann 2000; Osher and Santosa 2001; Wang et

al. 2003), as well as being applied to an extended variety of topology optimisation

problems. These include compliance mechanics (de Gournay, Allaire et al. 2008) and

design of microstructures for materials (Mei and Wang 2004; Wilkins, Challis et al.

2007; Challis, Roberts et al. 2008). These materials include those with negative

Poisson’s ratio (Wang and Wang 2005b), specific electromagnetic characteristics

(Zhou, Li et al. 2010; Zhou, Li et al. 2011) and negative permeability (Zhou, Li et al.

2011).

The name for the level set approach comes from the function describing the boundary of

structure (Challis 2010). In some domain , the level set of the scalar function

RR: 3 →ϕ is defined as:

{ }z,tt:ttS == ))(φ()()( xx (2.10)

Here is a constant value called iso-value, usually taken as zero in structural problems.

The design domain will be divided into 3 regions following this definition as

⎪⎩

⎪⎨⎧

Ω∉>Ω∂∈=Ω∈<

=xxx

x if 0 if 0 if 0

)(ϕ

(2.11)


31

Here the area covered by domain Ω is filled with material. Ω∂ defines the structural

boundary. The level set function is defined according to the center’s position of

elements ci in finite element modeling of the structure, where the elements are

differentiated as either solid or void elements. It can be expressed as (Challis 2010)

⎩⎨⎧>

=<=

otherwise 01 if 0

)( ei

xcϕ

(2.12)

The structural boundary Ω∂ changes during process of structural optimization as a

result of the level set function S(t) dynamically changing in time. So does the surface

develops as a result of the specified “speed vector” of level set surface at different

points. A so-called “Hamilton-Jacobi” type equation can be obtained via deriving

equation (2.10) as a function of time and applying the chain rule:

vϕϕϕ ∇−=∇−=∂∂

dtdx

t . (2.13)

This equation establishes the relation of the speed vector of the point on the surface to

the objective of optimization. Also the optimal structural boundary can be articulated as

a numerical solution to this partial differential equation on ϕ (Wang, Wang et al.

2003).

One advantage of the level set approach is that sharp interfaces between different

constituent phases of the structure can be obtained, which simplifies the interpretation

of boundary and manufacturing compared to other topology optimization methods such


32

as SIMP, which uses continuous variables (Burger and Osher 2005). On the other hand,

the classical formulation does not allow the systematic formulation of new holes in the

topology of structure, especially in two-dimensional cases (Allaire, Jouve et al. 2004).

As the level set method generally describes the propagation of interfaces with a defined

speed function, holes within existing shapes and away from the boundaries cannot be

initiated (Burger, Hackl et al. 2004).

Various solutions were proposed to overcome problems relating to the nucleation of

new holes in the structure. A large number of discrete holes could be introduced in the

initial design (Allaire, Jouve et al. 2004). The above-mentioned level set setting is

capable of merging or cancelling these holes and creating a structure with fewer holes in

following iterations. Still the level set method is not capable of creation of further holes

during the optimization process, but only in initial stage. This is the reason why the

number and location of the initial holes largely influence the final solution (Wang,

Wang et al. 2003; Allaire, Jouve et al. 2004).

Another method to solve the nucleation of holes is to apply q interpolation (Burger,

Hackl et al. 2004). The modified Hamilton-Jacobi differential equation that needs

solving will then have the following format:

wqt

−∇−=∂∂ vϕϕ

(2.14)

Where w is a positive weighting factor which determines the influence of the term q.

The determination of q and are involved with the sensitivity analysis of the


33

optimization objective function. The selected q is dependent on the problem at hand and

its weighting factor should be determined by the user as an initial parameter (Challis

2010) and therefore its successful application is highly dependent on previous

experiences. The introduction of additional constraints in the level set approach

involves further modification into the Hamilton-Jacobi differential equation (2.14)

through addition of extra weighted terms (Challis, Roberts et al. 2008). Thus,

successfully implanting the method in conjunction with extra constraints becomes

cumbersome in two-dimensional problems.

Generally, the level set approach is considered to be more mathematically complicated

and difficult to implement as a computational procedure, in contrast to materials

distribution approaches, the SIMP, ESO and BESO. This has prevented it from regular

application (Rozvany 2009). As for materials distribution approaches, due to their

mathematical simplicity, these methods have received greater attention besides being

more developed.

2.5 Bi-Directional Evolutionary Structural Optimization (BESO) The bi-directional evolutionary structural optimization (BESO) method was the next

generation product based on its 1.0 version of the evolutionary structural optimization

(ESO) method. Research on this area has been very active and extensive with many

algorithms proposed within the past fifteen years since the term “BESO” was first

introduced (Querin 1997; Querin et al. 1998). The various versions of former BESO

methods attempted to solve shape/topology optimization problems empirically and have


34

been unsuccessful in guaranteeing optima in the solutions (Rozvany 2001). Recently, an

advanced BESO approach which yields mesh-independent and convergent solutions

was proposed by Huang and Xie (2007). This was followed by proposal of a

mathematically established formulation for this BESO (Huang and Xie 2009) which

solves the stiffness optimization problem using soft-kill approach (i.e. replacing void

elements with soft material) that is shown to be equivalent to hard-kill (complete

removal of elements designated as void). Final optima can be ensured in this advanced

version of BESO by incorporating a more rigorous optimality criterion.

An overview of the BESO method is presented in following session with basic concepts

of the BESO method explained and formulations devised for the most simple and basic

situation which is periodic optimal design under given ratio with one single base

material.

Although in BESO method the most obvious advance than ESO is that it’s capable to

re-admit elements into the design domain, in previous versions (Querin 1997; Querin et

al. 1998; Yang et al. 1999; Zhu et al. 2007) it was still not capable of ensuring optima

(Rozvany 2001; Zhou and Rozvany 2001; Rozvany 2008). This was due to a number of

reasons from various aspects. Initially, the assumption of sensitivities for void elements

was not accurate which further more delivered inaccurate estimation on the change of

the optimization objective. In some previous versions, void elements around high-

performance solid elements were subject to heuristic re-admission, owing to insufficient

information on the absent elements. Also, non-convergence prevents the production of

reliable results. This shortcoming requires one to pick out the best solution among


35

several possible ones which can be very difficult and lose the point of optimization. At

last there is neither direct mathematical explanation for the approach nor indirect

demonstration to confirm that the design evolves towards an optimum. So as a result the

removal or admission of elements is quite intuitive driven.

Moreover, an essential feature expected from most of the popular topology optimization

techniques is mesh-independence. Mesh-dependence is considered a numerical

instability (Sigmund and Peterson 1998; Bendsøe and Sigmund 2003), which leads to

the occurrence of dense holes in the final design, usually called the checkerboard

pattern. These make further detailed designs unfeasible. Attempts to overcome this have

been made using algorithms such as perimeter control (Yang et al. 2003) and a

smoothing algorithm (Li et al. 2001). The filter scheme developed by Huang and Xie

(Huang and Xie 2007) is used in this present work. The filter scheme is able suppress

the checker-board patterns and work for mesh-independence, besides also acting as an

effective mechanism for extrapolating sensitivity numbers from solid elements to void

elements.

An improved and mathematically based BESO approach was proposed by Huang and

Xie (Huang and Xie 2007; Huang and Xie 2009) to overcome the difficulties described

above. In this improved BESO version, soft-kill of elements is applied instead of hard-

kill, which means when a solid element is replaced, it would be by soft element rather

than void element. The filter scheme makes up one of the basic algorithms in this new

version, which is able to guarantee final optima by applying such rigorous optimality

criteria. This improved BESO method has been applied to a variety of stiffness


36

optimization problems and proven to be capable to achieve mesh-independent and

convergent design results.

2.5.1. Hard-kill BESO

Following the ESO method which based on the idea of gradually removing inefficient

elements off from the finite element model of the structure, the method called “additive

evolutionary structural optimization” (AESO) has been introduced targeting generating

optimum structures with initial design of a minimum ground structure and gradually

adding elements to it (Querin, G.P.Steven et al. 1998; Querin, Steven et al. 2000). In

the AESO method, new elements would be added to the free edges of the most efficient

elements where the most efficient elements are selected by the standard of elements

with highest stress or sensitivity numbers (Querin, Steven et al. 2000). BESO, another

member in the ESO method family tree and yet the most developed one, was created

along the trend with added flexibility and stableness (Yang, Xie et al. 1999). In BESO

method elements can be added and/or removed in each iteration. The criteria for adding

or removing of elements are based on their effects on variation of objective functions.

The numbers of added or removed elements are controlled by two given parameters, the

inclusion ratio (IR) and rejection ratio (RR) respectively.

As mentioned before sensitivity numbers expresses such effects. In a classic BESO

project, the sensitivity numbers are calculated according to the results of structural

analysis for solid elements. While for void elements the sensitivity numbers are

calculated based on their nodal displacements, which are calculated by extrapolating the

nodal displacements of their surrounding solid elements. The method takes after


37

primarily by the ranking of elements in view of the extent of their sensitivities. An

element will be changing to solid if it is with higher sensitivities, or to void for those

who have lower sensitivity numbers. Like soft-kill BESO method, the quantities of

removed and added elements are treated with two separate criteria by adding a

constraint of volume-fraction changing ratio between two adjacent iterations.

The urge for an improved optimization method had been raised due to dissatisfaction of

earlier solutions. As mentioned earlier, the proposed optimization problem of solid-void

material distribution could not reach a universal solution because the elements sizes and

discretization meshes fatally influences this supposed optimization problem as

demonstrated by Bendsøe and Kikuchi (Bendsøe and Kikuchi 1988). Other

disadvantages of these earlier methods include that the numerical instability was not

handled well as well as computational efficiency was relatively low due to the

convergence problems (Rozvany 2009; Huang and Xie 2010a). Furthermore we need to

list out all possible topologies that are generated via various RR and IR in order to find

out the final best solution (Rozvany 2009; Huang and Xie 2010a).

Huang and Xie (Huang and Xie 2007) developed a new algorithm for the hard-kill

BESO in 2007. In this version they addressed several issues including a clearer

statement of the optimization problem and better solution against numerical instability

during the procedure (Huang and Xie 2010). Here it was supposed that the purpose of

the optimization was to find the stiffest structure with volume as additional constraint.

In this version of hard-kill BESO method, the optimization problem is stated as:


38

Minimize: f(x)=K (2.15.a)

Subject to: 0

1

* =−∑=

N

iii xVV

(2.15.b)

1or 0=ix . (2.15.c)

Here the xi is a design variable which indicates the absence or presence of an element in

the PBC, as 1 to be present and 0 to be absent. This show in BESO method an element

is considered as the smallest unit and could only be 1 or 0, in contrast to the design

variable xi in SIMP approach which is between 0 and 1.

In order to estimate the sensitivity numbers of void elements, Huang and Xie developed

a filtering scheme based on the following weighting equation (Huang and Xie 2007):

∑

∑

=

== N

jij

N

j

niij

i

w

αwα

1

1ˆ

(2.16)

In the above equation N stands for the total number of finite elements in current design

iα model and is the calculated sensitivity number of element i. The weight factor ijw is

expressed as:

⎩⎨⎧ <−

=otherwise 0

if minmin rrrrw ijijij

(2.17)

Here rij states the distance between the centres of element i and that of element j. The

filter radius rmin defines till how far a neighbouring element could have an impact on the


39

sensitivity of element i. Initially the sensitivity numbers of void elements were

temporarily assumed at zero, which would then be modified near the end of each

iteration through out the optimisation process according to the filtering scheme.

It would be followed by adding and removing of elements, both of which are based on

the rank of each element among all elements in the PBC. Those elements that have

lower sensitivity numbers would be switched to void elements; whereas for those

elements those have higher sensitivity numbers they would be designated to solid

elements instead. In a word, the filtering scheme is to rank each element by a number

calculated from its own sensitivity number and weight factors of surrounding elements

(Huang and Xie 2010).

The numerical instabilities in the original versions (Zhou and Rozvany 2001; Rozvany

2009) caused quite a few controversies. But by introducing the filtering scheme

described above (Huang and Xie 2010), more successful optimisations were archived

with less occurrence of so-said instabilities. With that being said, another significant

improvement made by Huang and Xie (Huang and Xie 2010) is the unified criteria for

adding and removing of elements. Volumetric constraint can then be carried out exactly

as a parameter by applying the criteria. Furthermore, this revised hard-kill BESO

method has much higher computational efficiency besides above-mentioned

improvements. It is as well for the reason that the removed elements would not be

engaged in following FEA, hence less elements to be considered better the time

consumption of the FEA (Huang and Xie 2010).


40

2.5.2. Soft-kill BESO

Rozvany pointed out that solid elements could only grow around or nearby existing

solid elements by applying above introduced Hard-kill BESO method (Rozvany 2001).

In some cases that may induce to failure in correcting the incorrect element rejection

(Zhou and Rozvany 2001; Zhu, Zhang et al. 2007). Besides that point, other findings

also specified that complete removal of void elements might also cause certain

dilemma, especially for multi-phase cases (Sigmund 2001; Zhu, Zhang et al. 2007;

Huang and Xie 2010a).

In attempt to solve these problems, Hinton and Sienz (Hinton and Sienz 1995) tested

another substitute approach based on ESO where the design domain is fully stressed and

pointing material properties to elements is bi-directional. Here bi-directional means

rather than complete removal of void elements a comparatively small density can be

assigned to the void elements, which is assumed possessing 106 times lower elastic

modulus than solid elements (Hinton and Sienz 1995). The bi-directional method allows

void elements to carry a stain value to stay in FEA hence they might be assigned as

solid elements along the optimisation process. In other words solid elements can grow

in any desired region of the structure instead of limited to around existing solid regions

(Rozvany 2001; Zhu, Zhang et al. 2007).

It’s worth mentioning another BESO method developed by Zhu, Zhang et al. (Zhu,

Zhang et al. 2007), which is sensitivity based. The innovation highlighted in this

method is to use orthotropic cellular microstructure (OCM) in place of void elements,


41

i.e. according to their sensitivity rankings elements would be assigned as OCM’s or

solid elements respectively. Here the OCM is defined as a microstructural system with

very low density. A filter scheme is applied to avoid numerical instability, which

controls the array of continued solid elements along each principal direction (Zhu,

Zhang et al. 2007). Though improvements have been made based on hard-kill BESO

nevertheless both methods experience some problems on convergence (Huang and Xie

2010).

It was until 2009 Huang and Xie published the soft-kill BESO method that the above-

mentioned hard-kill BESO limits were conquered. In this soft-kill BESO method when

an associated element is void, the design variable xi would be assigned to a relatively

small value xmin (e.g. 0.001), so as to keep such elements involved in continuing FEA

(Huang and Xie 2009). The optimisation problem here can be stated as

Minimize: f(x)=K (2.18.a)

Subject to: 0

1

* =−∑=

N

iii xVV

(2.18.b)

1or mini xx = (2.18.c)

The most common application is in stiffness optimisation where the sensitivity of

elements is based on the objective function f(x) with respect to design xi variable. Use

the case of Young’s modulus as effective property in stiffness optimisation as an


42

example. The objective function here is a function of the effective Young’s modulus of

PBC, which can be expressed though a power-law interpolation scheme (Bendsøe 1989)

pis xEE )(i )(x = (2.19)

Here E(s) stands for the Young’s modulus of the solid material. p is a penalty exponent

assigned manually. There is a similarity of results observed between this soft-kill BESO

method and the hard-kill one (Huang and Xie 2007, Huang and Xie 2010). As stated by

Huang and Xie (Huang and Xie 2010a), the sensitivity numbers of soft elements are

dependent on penalty exponent p. When p approaches infinity, the sensitivity numbers

of solid elements and “void” elements would become the elemental strain energy and

zero respectively, same as that of the hard-kill BESO method. Such consistency also

happens to the objective function. Hence the hard-kill BESO method can be seen as a

special case of the soft-kill BESO method with penalty exponent p→∞.


43

Chapter3TopologyoptimizationofCompositesforMaximisingEffectiveYoung'sModuli

In the case of composites design for optimal stiffness, the effective Young’s modulus

and Poisson’s ratio of the composite material are obtained through homogenization

theory. Single or multiple objectives are defined to maximize these properties separately


44

or in combination. Based on the sensitivity analysis of the objective function, a BESO

calculation is conducted to achieve optimized topology for the composite unit cell.

This chapter investigates the effect of Poisson’s ratio on composite materials. The study

is focused on composites containing two materials with different Poisson’s ratios,

especially when one of them is nearly incompressible, to reveal the role that Poisson’s

ratio plays in optimized composites. Two types of problems are used: maximizing E3

(where E1=E2) or maximizing E1, E2, and E3 together (where E1=E2=E3). The composite

is modelled as a microstructure in a periodic unit cell. A combined objective function is

defined in terms of the largest Young’s modulus of the composite. The overall

optimization in this study is based on the bi-directional evolutionary structural

optimization (BESO) method.

Several research questions to be answered, as follow:

1) Is it possible to build a composite with higher Young’s moduli than its base

materials? How much higher can we achieve?

2) How to develop a Fortran code for this composites optimization?

3) Is it possible to find the best volume fraction automatically?

4) What is the suitable interpolation scheme and initial design?


45

5) When expanding the range of Poisson’s ratio to (-1, 0), will this program be

applicable? And how will the negative Poisson’s ratio affect the optimization

procedure and results?

6) Is there any limitation for this method to be applicable and why?

3.1. Methodology

3.1.1 Optimization problem statement

The optimisation problem can be stated as follows:

Maximize:)(

31

321 EEEf ++=, (3.1a)

Subject to:0

1

* =−∑=

N

eee xVV

,

minxxe = or 1, (3.1b)

Where E1, E2 and E3 are the effective Young’s moduli of the composite. Equation (3.1a)

stated that all the three effective moduli are to be maximized. Alternatively, one can

choose to maximize a single modulus, that is

Maximize: f =E3. (3.2)


46

3.1.2 Optimization with optimum volume to be solved

In BESO, the above problems are solved by iteratively searching the optimum

according to the element sensitivity, denoted as αε. For problem as defined in Eq. 2, the

optimum volume is estimated at each iteration according to the element sensitivity. The

problem statement of this case can be generally stated as ‘to find the value of x (volume)

at which the function f(x) has its maximum. When f(x) is the bulk modulus K, the

solution is simple, that is x =1 and f(x)=K2. When f(x) is E as in this case, x is to be

solved by incorporating an addition algorithm in BESO.

The methodology is as follows. The volume target is estimated at each iteration

according to the element sensitivity. Assume that the topology has totally NE element,

and at iteration i, the number of element of materials 1 and 2 are NE1 and NE2,

respectively.

Check the element sensitivity eα , and

l Sum up the number of elements of material 2 which has 0<eα , denoted as −2NE .

l Sum up the number of elements of material 1 which has 0>eα , denoted as +1NE .

Then the target volume is calculated as

NENENENEVt

+− +−= 122

(3.3)


47

An example is shown in Figure 3.1. The two base materials have Young’s modulus 1E =

2E =1 and Poisson’s ratios of 1v =0.2 and 2v =0.48. Since the models possess three

orthogonal planes of symmetry, they can be simplified to one-eighth models. The

topologies in Figure 3.1 (a) & (b) have a volume constraint V*=0.4705 and an optimum

volume V=0.4333, respectively. Their effective moduli have increased by

approximately 3% from the base value.

The iteration history of the case of optimum volume is shown in Figure 3.2. It is seen

that the actual volume gradually approaches the target volume and the two finally

converges. At iteration 85, the actual volume and target volume are 0.4333 and 0.4334,

respectively, where E reaches the maximum (1.037317).

(a) VolumeconstraintV*=0.4705,

E1=E2=E3=1.037209

(b) OptimumvolumeV=0.4333,

E1=E2=E3=1.037317

Figure 3.1 Optimal topologies of maximizing )(31

321 EEEf ++=

Mat 1, v=0.2

Mat 2, v=0.48


48

0.0

0.2

0.4

0.6

0.8

1.0

1.000

1.005

1.010

1.015

1.020

1.025

1.030

1.035

1.040

0 20 40 60 80 100

Vol

ume

V

Stif

fnes

s E

Iteration History

E

Target volume

Actual volume

Figure 3.2 Iteration histories of effective modulus and optimum volume

3.1.3 Verification on sensitivity analysis

One essential procedure of the BESO algorithm is to calculate the element sensitivity.

This part is to verify that for problems dealing with different Poisson’s ratio, such

calculation is conducted correctly in the BESO code.

There are three aspects in the calculation, i.e.

a) Assumption of interpolation function.

b) Implementing the equation of sensitivity.

c) Implementing the filter procedure.

They are investigated/verified as follows (reversely ordered).

c) Verification of filter procedure


49

This procedure is common to all BESO. The current code has been used to produce

correct results for ( a. Radman, Huang, & Xie, 2012; A. Radman, 2013; Yang, 2012).

The same subroutine of filter procedure is used for results of different Poisson’s ratio.

Therefore, it is concluded that the code for filter is correct.

b) Verification of equation of sensitivity

In doing so, sensitivity number

cba

e

e

ee

e

ee xxv

xvxxE

xEx

MMM

EEE

++=

∂∂

∂∂+

∂∂

∂∂=

∂∂ )(

)()(

)(

(3.4)

is calculated manually using Excel, and then is compared to the results from codes

(Subroutine cal_dEmatrix_dx). Three cases are studied. The manual calculation results

are shown from one cell. The results from BESO are output to file ‘temp_CalE.txt’, and

extracted to another cell. They have matched well in all three cases and therefore the

code is verified.

a) Assumption of interpolation function

It has been discussed before that for BESO of different Poisson’s ratio, it is difficult to

determine whether an interpolation function is appropriate. An interpolation function is

to reflect a certain physical phenomenon. For example, for SIMP with penalty >3, the

interpolation of E can be used to realize the HS bounds(Bendsùe & Sigmund, 1999).


50

Up to now it can be said that the interpolation of Poisson’s Ratio is speculative. In

(Bendsùe & Sigmund, 1999), Sigmund has suggested an interpolation as

It is also pointed out that ‘the low volume fraction limit has a Poisson ratio equal to

1/3’, which can be understood that v>=1/3 when x=xmin. This may suggest that there

exists a range on which an interpolation is held valid.

If this argument is correct, it may help us understand why BESO not working ‘properly’

when, say v2>0.49 and when v1<0.3. The question we can ask can be:

1) Could it be that the interpolation we’ve assumed is not valid beyond certain point,

and this point happens to be around 0.49.

Also, it is interesting to note that in case study, BESO has worked well when v2>0.495

but a higher v1 withv1=.0.3, then the question can also be:

2) Could it be that the interpolation assumed is not valid when the two Poisson’s ratios

are “too different”?

It is noted that BESO results seemed incorrect with v2>0.49 and v1<0.3. In this part,

possible causes to this have been investigated. Errors in codes with respect to sensitivity


51

analysis have been ruled out. One other possible cause is the choice of interpolation

function. It is likely that although hard to be quantified, an interpolation function may

be only valid within a certain range.

3.1.4 Compare material interpolation schemes

Two material interpolation schemes are proposed, i.e.

1) Interpolate the E matrix

21)1( EEE pe

pe xx +−= (3.6)

2) Interpolate the E and v respectively

21)1()( ExExxE EE pe

pee +−= (3.7a)

21)1()( vxvxxv vv pe

pee +−= (3.7b)

For using E matrix interpolation, it is noticed that the topology changes rather random

between iterations. This may suggest that the searching is not in the right path. Changin

g the interpolation from E matrix to E and v interpolation, the change between iterations

becomes gradual, and reasonable results are obtained.


52

Table 3.1 Compare material interpolation schemes through iteration histories

Interpolate the E matrix Interpolate the E and ν respectively

Iteration 000

Iteration 001

Iteration 002

Iteration 003


53

Another two interpolation schemes are investigated as well, i.e.

1) SIMPinterpolation:

E(xe ) = E2xe

p + E1(1− xe

p ) (3.8)

2) Qinterpolation:

E(xe ) = E1 +xe(E2 − E1)1+ q(1− xe )

(3.9)

When both base materials possess positive Poisson’s ratio and maximizing E3:

penalty=1 got highest E3 and comparably lower E1 and E2; compare SIMP and Q

interpolation, SIMP interpolation focus better on E3 maximization. In future studies,

SIMP interpolation with penalty p=1 will be used when maximizing E3 (where E1=E2).

The topology presents a constant pattern of base material 2 re-distributed along the

strongest axis while the overall stiffness is optimized. It was noticed that case study on

v2=0.49 ended up with incorrect result. The current method is not compatible when

v2≥0.49.

Table 3.2 Base materials ---- Eb1=1, v1=0.1, Eb2=1, v2=0.48

Interpolation Penalty Optimum

volume E1 E2 E3 Topology

Q 1 0.4941 1.04268 1.04268 1.11017 (a)

Q 2 0.4919 1.05242 1.05242 1.09311 (b)

Q 3 0.4831 1.05950 1.05950 1.08175 (c)

SIMP 1 0.4269 1.04096 1.04096 1.11545 (d)


54

SIMP 2 0.4109 1.06498 1.06498 1.07544 (e)

SIMP 3 0.4109 1.06498 1.06498 1.07544 (f)

(a)

(b)

(c)

(d)

(e)

(f)


55

Table 3.3 Base materials ---- Eb1=1, v1=0.1, Eb2=1, v2=0.49

Interpolation Penalty Optimum volume E1 E2 E3 Topology

Q 1 0.4320 1.05931 1.05931 1.09832 (g)

SIMP 1 0.4298 1.04345 1.04345 1.12355 (h)

(g)

(h)

3.1.5 Compare Initial designs

Two initial designs are used, as shown in Figure 3.3.

l Initial A: with v=0.48 as the small inclusion. This is also used in the above results

in part 3.1.2.

l Initial B: with v=0.2 as the small inclusion.


56

Initial A Initial B

Figure 3.3. Initial design and corresponding optimum topology.

Table 3.4 Effective Young’s modulus for different initial designs

Case Volume V E

Initial A 0.4333 1.037317

Initial B 0.4767 1.035530

The iteration history of initial A is as previously shown in Figure 3.2, and the initial B

now shown in Figure 3.3. The initial volume is high and it is gradually reduced to

approach the target. The results are presented in Table 3.4. The values of E and volume

for two initial designs are different, which is unexpected. Also comparing with the

Volume V=0.01 Volume V=0.99


57

previous cases, E value of Initial B is lower. This may suggest that the result is

incorrect.

Figure 3.4. Iteration history for initial design B

Here examine the possible explanation for the low value. Combine the iteration history

for initial A (Figure 3.2) and initial B (Figure 3.4). Now use the volume instead of

iteration as the x- axis. Initial A starts from the lower volume and initial B starts from

high volume. There is a gap between the two respective maximum points. How the

optimum E changes between the gaps is questionable. One possibility is that it may

have some local peaks, as sketched in the close-up of the circled area. After Initial B

reaches the local peak B, the search will terminate and be unable to follow the drop


58

before reaching local peak C, therefore will miss the peak A which is considered as the

global optimum.

This kind of multi-valued objective function, if possible, can be a relative new

phenomenon. From mathematical point of view, to numerically search the maximum of

a multi-valued function, it is possible to be ‘trapped’ in a local maximum, which is

closest to the searching starting point. Unfortunately there might not be simple solution

and more investigation of the nature of the problem may be needed.

Figure 3.5. Searching path from two different initial design

3.2 Case study with negative Poisson’s ratio

When one of the base materials possesses negative Poisson’s ratio, initial design plays

an important role for achieving correct results. When maximizing E3 only, it’s better to

A B

Initial B Searching from V=1.

Initial A Searching from V=0.

A

B C


59

use material with negative Poisson’s ratio as the small inclusion for a better result, and

the composite turned out in sandwich shape. When maximizing E1+E2+E3, we can

remain using material with positive Poisson’s ratio as the small inclusion.

• Initial design A: Base materials (Eb1=1, v1= -0.9, Eb2=1, v2=0.4)

• Initial design B: Base materials (Eb1=1, v1= 0.4, Eb2=1, v2=-0.9)

Table 3.5 base materials with negative Poisson’s ratio

(a)

(b)

In the above case studies, both base materials have positive Poisson’s ratios. Here we

consider the Poisson’s ratio of one of the base materials is negative, e.g. v=-0.9.

Initial design Maximize Optimum

volume E1 E2 E3 Topology

B E3 0.6436 1.64077 1.64077 3.68979 (a)

A E1+E2+E3 0.4360 3.26924 3.26924 3.26924 (b)


60

* Initial design A base materials E1=1, ν1= -0.9, E2=1, ν2=0.4. ** Initial design B base materials E1=1, ν1= 0.4, E2=1, ν2=-0.9.

As shown in Table 2, when maximizing E1+E2+E3, the topology has a spherical

inclusion of positive Poisson’s ratio. When maximizing E3 only, the topology has a

sandwich like configuration with the material of negative Poisson’s ratio being in the

middle. The effective moduli are approximately three times of that of the base material

which is a significant increase.

Table 3.6 Topologies from base material of negative Poisson’s ratio

Maximize Initial design Optimum topology Volume

fraction Young’s moduli

E1+E2+E3

0.4360 E1:3.26924 E2:3.26924 E3:3.26924

E3

0.6436 E1:1.64077 E2:1.64077 E3:3.68979

B**

A*


61

3.3 Concluding remarks

In this chapter, the author presents the BESO method in designing composite materials

of maximum stiffness. The composite consists of base materials of different Poisson’s

ratios. The moduli of the composite are found to be higher than the base materials.

When one of the base materials has negative Poisson’s ratio, the increase in the moduli

is very significant.


62

Chapter4TopologyoptimisationofcompositesforMaximisingBulkorShearModulus

Compared with single-phase material, or say solid material, composite material may

have advantages in many aspects, such as physical, mechanical and thermal properties.

The physical properties of composite material are different when changing the material


63

distribution within its microstructure. And the method of changing spatial distribution

of microstructure/material in order to achieve aimed properties is called topology

optimization method for microstructures/materials. In this chapter, bi-directional

evolutionary structural optimization (BESO) will be applied to design the periodic

composite microstructures, which consist of two solid phases. The objective function is

to optimise a single physical property including maximising bulk modulus and shear

modulus. Here in this study the weight of the structure is not considered as an aim of

design like in porous microstructures, but still participates in design process by imposed

as volume constraints.

Here we define the smallest repeating unit of composite microstructure to be periodic

base cell (PBC). Its dimensions are required to be small enough compared with overall

dimensions of the whole design body. For such circumstance, homogenization theory

can be introduced in order to establish the relationship of material properties in between

the whole design body that is in macrostructure scope and the periodic base cell (PBC)

that is in microstructure scope. The PBC is then discretised into finite element model

under periodic boundary conditions. Next, the finite element analysis is carried out to

observe the performance of each designed material distribution in PBC.

Another concept to be introduced is sensitivity number, as well as related sensitivity

analysis. Sensitivity number is known as the effect of an individual element on pursued

objective function. And sensitivity analysis is the calculation so as to get sensitivity

number (Haug, Choi et al. 1986; Huang and Xie 2010a). By ranking the sensitivity


64

numbers of each individual elements in PBC from high to low, we can iteratively decide

whether each element property to be solid ( 1=ix ) or void ( 001.0min =x ) in BESO

method. Thus the PBC’s topology is gradually optimized towards satisfying both

convergent criteria and volume constraint along with granting solid or void property to

each element by iteration (Huang and Xie 2010a).

The details of the process will be deployed in this chapter, followed by presenting

numerical examples. The results of the topology optimization are then verified with

known analytical bounds on material properties.

4.1. Methodology

4.1.1 Problem statement of periodic material topology optimization

Both bulk modulus K and shear modulus G are important figures marking the stiffness

of elastic materials. In this section we are stating the design problem of periodic two-

phase composite materials with maximum effective bulk modulus or shear modulus

subject to prescribed volume constraint. Therefore the topology optimization problem is

to find the appropriate distributing of two solid phases within the PBC subject to a

prescribed volume fraction of each solid phase. Such an optimization problem can be

described as:

Maximize: Kf =)(x or G

Subject to: 01

* =−∑=

N

iii xVV (4.1)


65

1orminxxi =

Where *V and iV are the volumes of the design domain (PBC) and each individual

element respectively. N is the total number of elements within the PBC. And ix is a

binary design variable which represents the density of i th element.

Bulk modulus K shows the stiffness of a certain material under uniform pressure and it

can be calculated from component of material effective elasticity matrix ( HiiD ). Under

3D circumstances bulk modulus can be expressed as

K = 19(D11

H + D12H + D13

H + D21H + D22

H + D23H + D31

H + D32H + D33

H ) (4.2)

Shear modulus describes material response under shear strains and it can be calculated

from component of material effective elasticity matrix ( HiiD ) as well. When in 3D cases

shear modulus can be stated as

( ) 31

665544HHH DDDG ++= (4.3)

As only orthotropic cellular material with cubic symmetry in 3D cases are considered in

this chapter, we can also have the following relationships:

332211 DDD == , (4.4)

32232112 DDDD === 1331 DD ==


66

665544 DDD ==

4.1.2 Topology optimization through BESO method

Computational topology optimizations are all based on the concept of material

distribution, which is to assign material properties of base materials to each elements of

the finite element model of the PBC, in order to get an optimized design function. As

one of the computational topology optimization methods, the developed BESO method,

established by Huang and Xie (Huang and Xie 2010a), decides material distribution

using sensitivity analysis, which is done by calculating and ranking the impact of each

individual element on a certain object function. As mentioned before such calculation

gives each element a sensitivity number. To be more specified the calculation is to

derive a certain objective function with respect to the binary design variable ix of the

thi element in PBC, and result in sensitivity number iα as shown below in equation

(4.5).

ii x

f∂∂= )(xα (4.5)

Following the sensitivity analysis is optimization process. The idea of optimization

process is to assign solid material to more critical or say influential elements (Huang

and Xie 2009a; Huang and Xie 2010a). Here critical elements are selected after ranking

the sensitivity numbers of all elements in design domain, and those of higher sensitivity

numbers are critical elements. To carry this out in the practice of BESO method,

elements with higher sensitivity numbers iα will be assigned as solid elements ( 1=ix ).


67

And on the other hand elements with lower sensitivity numbers iα will be assigned as

void elements ( minxxi = ) (Huang and Xie 2010a). In another word the binary design

variable ix represents the density of i th element as mentioned in (4.1). When adapted

to design case of two solid base materials, the one possessing higher required property

can be defined as “solid”, and the other one is comparatively “void”. Considering the

lower bound ( 001.0min =x ) rather than 0 for the design variable, helps to capture the

effect of “void” elements in the analysis since the sensitivity of “void” elements could

also be calculated. As a result such adjustment cuts numerical problems by helping

avoiding complete removal of elements and brings benefit by growing solid elements in

new desired regions of the structure (Huang and Xie 2009a; Huang and Xie 2010a).

As mentioned in Chapter 3, when both base materials are isotopic, the Young’s modulus

of each element in PBC can be interpolated under a power-law scheme as below as a

function of the design variable marking element density:

E(xi ) = E2xip + E1(1− xi

p ) (4.6)

Here E1 and E2 denote the Young’s moduli of both base materials. ix denotes the

element density of the thi element and p is the penalty exponent. Huang and Xie

(Huang and Xie 2009a; Huang and Xie 2010a) have studied the function and impaction

of penalty exponent p . When using penalty exponent 1=p , the material interpolation

scheme is linear. This was tested as the simplest form, in combination with no filtering

scheme, and consequently outcomes of these cases showed numerical instability while


68

solutions were not necessarily an optimum. Whereas when using penalty exponent

1>p (usually choosing 3=p ), in combination with filtering scheme to reduce

numerical instabilities, results show better convergence of the procedure with more

distinct solid and void regions in the structure (Huang and Xie 2009a; Huang and Xie

2010a). That’s because filtering scheme is functioning as “damper” to reduce the

differences between sensitivity numbers of solid and void elements. So when given a

proper penalty exponent (eg. 3=p ), solid and void regions can grow and build up in the

whole structure, and finally converge to an optimized result.

In order to tailor the BESO method for the design purpose of this study, we need to

have a look into the evaluation of effective properties of the PBC and the derivation of

sensitivity numbers for each element within the PBC, which will be unfolded in next

section.

4.1.3. Homogenization and Sensitivity Analysis

As mentioned ahead, the spatial distribution of base materials within the PBC has an

impact on the overall effective properties of the composite microstructure. So in order

to design a composite microstructure, we must first figure out how to calculate the

overall effective properties of the composite based on the spatial distribution of base

materials within the PBC. In this study the designed material/structure is composed of

periodic base cells with all three dimensions much smaller than material’s/ structure’s

macroscopic length scale but larger than atomic length scale. Hence the effective

properties of the designed material/structure can be evaluated by analysing the periodic


69

base cell with the help of homogenisation theory (Bendsøe and Kikuchi 1988; Hassani

and Hinton 1998a; Hassani and Hinton 1998b). According to homogenisation theory,

the elasticity tensor of the composite can be expressed as below in (4.7) based on

material distribution in the design domain of the periodic base cell Ω.

∫Ω

Ω−= dEY

E klpq

klpqijpq

Hijkl )~(1 εε (4.7)

Here Y represents the volume of the periodic base cell Ω . klpqε defines linearly

independent test strain fields which in 3D problems consists of 6 fields respectively. For

the convenience of calculation and analysis the test strain fields are usually defined as

unit strains along principal directions. klmnε~ defines the strain fields induced by the test

strains. klmnε~ can be calculated from the following equation:

Ω=Ω∫ ∫Ω Ω

dvEdvE klpqijijpq

klpqijijpq εεεε )(~)( (4.8)

Here )(1 Ω∈ perHv represents the Y-periodic tolerable displacement field. Equation (4.8)

is the weak form of the standard elasticity equation that applies to PBCs with periodic

boundary conditions. Also (4.8) can be solved by finite element analysis of periodic

base cell according to linearly independent test strain fields klpqε . With the help of

interpolation under power-law scheme introduced in equation (4.6), in combination with

(4.7) and (4.8), the sensitivity number of homogenized elasticity tensor, which is


70

derived as a function of the binary design variable ix , is expressed as below in (4.9) by

the adjoin variable method (Haug, Choi et al. 1986).

∫Ω

Ω−−∂

∂=

∂∂

dxE

YxE ij

rsijrs

klpq

klpq

i

pqrs

i

Hijkl )~)(~(1 εεεε (4.9)

Considering the test strain fields as unit strains fields on the periodic base cell, the

homogenized elasticity matrix HD of designed materials/structures can be expressed in

vector form as:

∫ −=Y

H dYY

))((1)( BuIxDux,D (4.10)

In (4.10) u represents the displacement field. It is calculated from equivalent forces

that cause unit strains and finite element analysis of the PBC under periodic boundary

conditions. I is the unit matrix. B is the strain-displacement matrix. The sensitivity

number of homogenized elasticity matrix, in the form of derivation of HD as a function

of the binary design variables ix , can be expressed as:

∫ −∂∂−=

∂∂

Y i

T

i

H

dYxYx

)()(1 BuIDBuID (4.11)

Combining equations (4.11) with equations (4.2) and (4.3), we can have the sensitivity

numbers of both objective functions.


71

Up to date on the BESO method, it is usually assumed that the isotropic base material

changes only in the Young’s modulus, not the Poisson’s ratio. Literatures on the

sensitivity with respect to Poisson’s ratio are limited. In this study, base materials can

have not only different Young’s moduli but also their own Poisson’s ratio. To apply the

above analysis to the specific design case of this chapter, below is the derivation for the

sensitivity number of the homogenized elasticity tenser with base materials possessing

different Poisson’s ratios.

For a PBC consisting of different base materials, the elastic tensor is

∑ ∫∑==

⎟⎠⎞⎜

⎝⎛ −−==

NE

e eejj

Ti

TiY

NE

e

eij

Hij dY

YQE

e1

00

1)()(1 εεEεε (4.12)

(i, j=1 to 6 for 3D),

and ejjTi

TiY

eij dYY

Qe

)()(1 00 εεEεε −−= ∫

Where E is the elastic matrix of the base material, 0iε is the i th unit test strain field and

iε is the corresponding induced strain field. eijQ represents the element mutual energy.

The sensitivity of HijE is

dYxYx

Ejj

e

Ti

TiY

e

Hij

e)()(1 00 εεEεε −

∂∂−=

∂∂

∫ (4.13)


72

The elasticity matrix of the isotropic base material E is a function of the Young’s

modulus E and Poisson’s ratio v. Assume that there are two base materials, with E1 and

v1, E2 and v2. E can interpolated as

21)1()( ExExxE pe

pee +−= , (4.14a)

and it follows that

at min0 xxe ≈= ,

1)( ExE e ≈ (4.14b)

at 1=ex ,

2)( ExE e = (4.14c)

)()(12

1 EEpxxxE p

ee

e −=∂

∂ −

(4.14d)

also assume that

21 EcE e= (4.14e)

The interpolation of Poisson’s ratio had not been well studied. A quick search of

literature finds (Bendsùe & Sigmund, 1999) which only involves one base material. The


73

physical meaning associated with Poisson’s ratio interpolation is not clear yet. Here, it’s

assumed that Poisson’s ratio uses the same power law interpolation as the Young’s

modulus, i.e.

21)1()( vxvxxv pe

pee +−= (4.15a)

at min0 xxe ≈= ,

1)( vxvE e ≈ (4.15b)

at 1=ex ,

2)( vxv e = (4.15c)

)()(

121 vvpx

xxv p

ee

e −=∂

∂ −

(4.15d)

also assume

21 vcv v= (4.15e)

The elasticity matrix of 3D isotropic material


74

)())(21))((1(

)()(2100000

0)(21000000)(21000000)(1)()(000)()(1)(000)()()(1

))(21))((1()()(

eee

e

e

e

e

eee

eee

eee

ee

ee

xxvxv

xE

xvxv

xvxvxvxvxvxvxvxvxvxv

xvxvxE

x

B

EE

−+=

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

−−

−+==

(4.16)

Also define

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

−−

=∂

∂=

200000020000002000000111000111000111

)(

e

e

xxBC

, (4.17a)

ejjTi

TiY

eij dYY

Re

)()(1 00 εεCεε −−= ∫ (4.17b)

The 1st order derivative of the elasticity matrix is

cba

e

e

ee

e

ee xxv

xvxxE

xEx

MMM

EEE

++=

∂∂

∂∂+

∂∂

∂∂=

∂∂ )(

)()(

)(

(4.18)

Calculate Ma :


75

)())(21))((1(

)(

)()())(21))((1(

1)()(

121

xexvxv

EEpx

xxExe

xvxvxxE

xE

ee

pe

e

e

eee

e

ea

B

BEM

−+−=

∂∂

−+=

∂∂

∂∂=

−

(4.19)

at min0 xxe ≈= ,

11

11

min

minminmin

11min

minminmin

111

min

)11(

)())(21))((1(

)11(

)())(21))((1(

)1()(

)(

E

E

B

BEM

r

pxc

xxvxv

Epx

c

xxvxv

EEc

px

xxE

xE

p

E

p

E

E

p

e

e

ea

=

−=

−+−=

−+

−=

∂∂

∂∂=

−

−

−

(4.19a)

at 1=ex ,

22

2

2

12

)1(

)1())1(21))(1(1(

)1(

)1())1(21))(1(1()()(

)(

EE

B

BEM

rpc

vvE

pc

vvEcEp

xxE

xE

E

E

E

e

e

ea

=−=

−+−=

−+−=

∂∂

∂∂=

(4.19b)

Calculate Mb and Mc:


76

e

e

e

e

ee

e

e

ee

ee

e

ee

e

cbe

e

e

xxv

xx

xvxvxE

xxv

xxvxv

xExvxv

xvxxv

xv

∂∂

∂∂

−++

∂∂

−+−++

+=∂

∂∂∂

)()())(21))((1((

)()()(

))(21))((1(()(

))(21))((1(())(41(

)()(

BB

MME

(4.20)

e

ee

ee

e

ee

eb x

xvx

xvxvxE

xvxvxv

∂∂

−+−++

=)(

)())(21))((1((

)())(21))((1((

))(41(BM

(4.20a)

e

e

e

e

ee

ec x

xvxx

xvxvxE

∂∂

∂∂

−+= )()(

))(21))((1(()( BM

(4.20b)

at min0 xxe ≈= ,

11

111

min11

1

11

min111

1

11

1

)11()21)(1((

)41(

)11())21)(1(()21)(1((

)41(

E

E

BM

s

vc

pxvv

v

vc

pxvv

Evv

v

v

v

p

pb

=

−−+

+−=

−−+−+

+=

−

−

(4.21a)

CCM 111

11

1min )21)(1((

)11( tvv

Evc

pxv

pc =

−+−= −

(4.21b)

at 1=ex ,


77

22

2222

2

2222

1

22

2

)1()21)(1((

)41(

)1())21)(1(()21)(1((

)41(

E

E

BM

s

vcpvv

v

vcpvv

Evv

v

v

vb

=

−−+

+−=

−−+−+

+=

(4.22a)

CCM 222

22 )21)(1((

)1( tvv

Evcp vc =−+

−=

(4.22b)

Substitute Ma, Mb and Mc to (4.18) then to (4.13) we can have

dYYx

Ejjcba

Ti

TiY

e

Hij

e))()((1 00 εεMMMεε −++−=

∂∂

∫ (4.23)

at min0 xxe ≈= ,

eij

eij

eij

jjTi

TiY

e

Hij

RtQsQr

dYtsrYx

Ee

111

011111

0 ))()((1

++=

−++−=∂∂

∫ εεCEEεε

(4.23a)

at 1=ex ,

eij

eij

eij

jjTi

TiY

e

Hij

RtQsQr

dYtsrYx

Ee

222

011111

0 ))()((1

++=

−++−=∂∂

∫ εεCEEεε

(4.23b)


78

It’s worth notice that when considering two base materials 21 EE ≠ but common Poisso

n’s ratio 21 vv ≠ , the sensitivity based on (4.23) can be simplified as Mb and Mc being ze

ro matrix.

For cases of 21 EE ≠ and 21 vv ≠ , second order derivative is not taken into consideration

in (4.18) i.e.

e

e

e

e

ee

e

e

ee

e

ee

xxv

xxE

xvxE

xxv

xvxxE

xEx

∂∂

∂∂

∂∂∂

+∂

∂∂∂+

∂∂

∂∂=

∂∂

)()()()(2

1

)()(

)()(2E

EEE

, (4.24)

as the analytical expression for the last term can be very complicated and its impact on

sensitivity number is negligible.

4.1.4. Numerical instabilities and filtering scheme

There are two types of commonly seen numerical instabilities in topology optimization:

mesh dependency and checkerboard patterns. Mesh dependency is a type of instabilities

that result in structures of different effective properties by imposing different mesh sizes

in modeling of the structure. While the checkerboard problem is another type of

instabilities that is mainly caused by numerical errors and emerges when structural

modeling uses low-order finite elements. It appears in the pattern of elements in some

regions of the PBC switching between solid and void by iteration, and may block the

optimization process from convergence. Study by Sigmund and Petersson (Sigmund


79

and Petersson 1998) shows that by imposing restriction on related design variables both

problems can be avoided to a large extent.

In BESO method both mesh dependency and checkerboard patterns can be solved at the

same time by applying a filtering scheme that is inspired by a related technique used in

image processing (Huang and Xie 2007; Huang and Xie 2010a). The idea of the

filtering scheme is to use the average sensitivity number of one element itself and its

neighboring elements instead of the sensitivity number of only that element (Huang and

Xie 2007; Huang and Xie 2010a). This filtering scheme cuts solid regions with

dimensions smaller than a certain length-scale to appear in the generated topologies by

working as a low-pass filter. After applying the filtering scheme sensitivity number of

each element can be updated by the following equation:

∑

∑

=

== M

jij

N

jjij

i

w

w

1

1ˆα

α (4.25)

Here iα̂ is the updated sensitivity number. ijw is the weight factor. It can be expressed

as:

⎩⎨⎧

≥<−

=min

minmin

for0for

rrrrrr

wij

ijijij (4.26)

Here minr is the filter radius, which will be given as a specific number in design. ijr

represents the distance between the center of element i and j.


80

In order to help the optimization process reaching a convergent solution under

reasonably fine mesh density, iα̂ needs to be further modified by averaging itself with

its value from previous iteration (Huang and Xie 2007). This further modified

sensitivity number of each elements can be expressed as:

⎪⎩

⎪⎨⎧

+

== − otherwise )ˆˆ(

21

if ˆ~

1ti

ti

ki

i

1k

αα

αα (4.27)

Here superscript t represents current iteration number.

The new updated average sensitivity number iα~ replaces iα̂ from equation (4.25) as the

final sensitivity number of each element. It is used in ranking of all sensitivity numbers

and being recorded to be used in the sensitivity analysis of next iteration. The purpose

of averaging the sensitivity numbers with its iteration histories is to restrain the irregular

swinging of designed regions. The capability of the above filtering procedure for

stabilizing the optimization process is supported by large numbers of numerical

examples (Huang and Xie 2010a).

4.1.5. BESO Procedure

Drawing a summary to the previous sections of this chapter, a whole procedure for

BESO method for maximum bulk or shear modulus can be briefly expressed in the

following steps.


81

Step 1: Define the BESO parameters, such as prescribed volume fraction *V , filter

radius minr , penalty exponent p (normally )3=p ; evolutionary ratio ER and

mesh density.

Step 2: Create a finite element model for the PBC.

Step 3: Discretise the PBC using FE meshing with periodic boundary condition and unit

strain field. (In 3D problems 6 cases of loading and boundary conditions are

necessary.)

Step 4: Perform finite element analysis (FEA) on the current design and extract the

induced strain fields.

Step 5: Calculate the sensitivity number of each element for the design objective

function by combining equation (4.11) with equation (4.2) and (4.3).

Step 6: Perform the filtering scheme in the PBC design domain using equations (4.25)

and (4.26), and average sensitivity numbers with their historical information

using equation (4.27). Rank the sensitivity numbers of all elements.

Step 7: Define the target volume fraction of stronger elements for the next iteration. If

the current volume fraction of stronger elements tV is larger than the prescribed

volume *V , the target volume for next iteration is calculated


82

as )),1(max( *VERVV t1t −=+ . Also on the other hand define the target volume

fraction of weaker elements for the next iteration.

Step 8: Determine threshold sensitivity thα for element removal/re-admission. The

threshold sensitivity represents the breaking point on the ranking of sensitivity

numbers of all elements. Elements with a sensitivity number higher than thα

should have a total volume as the target volume fraction of stronger elements

determined in previous step.

Step 9: Redefine the material properties to all elements. For the elements whose

sensitivities are above the threshold ( thi αα ≥ ), they are defined as stronger

elements ( 1=ix ); and vice versa.

Step 10: Check if the remaining total volume of stronger elements meets the prescribed

volume fraction and if the optimization process satisfies the convergence

criterion.

Step 11: Repeat steps 2 to 10 until both the prescribed volume fraction and the

convergence criterion are satisfied.

The above-mentioned convergence criterion is defined by changes of the effective value

of the objective function (K or G ) over iterations, as seen in equation (4.28).


83

τη

θ

≤−

∑

∑

=

+−

=

+−−+−

1

1

1

11 )(

i

it

i

iNtit

f

ff (4.28)

Here f represents the effective value of the objective function. θ stands for the

summation upper bound. τ represents the prescribed convergence tolerance. θ is

usually set to 5 and τ is usually set to 0.1%. By applying these numbers to (4.28), it

can be explained as convergence is reached under the condition that the changes of the

effective value of the objective function over the last 10 iterations is equal to or less

than 0.1%.

4.2. Results and Discussion

4.2.1 Composites with maximum bulk modulus

Selected case study shown in this section use a cubic design domain, which is

discretized into 30×30×30 8-node brick elements. Each element is with dimensions 1

unit × 1 unit × 1 unit. Three sets of samples are studied, representing three assortments

of Poison’s rations of base materials. The optimization objective is to achieve maximum

bulk modulus, subject to volume fraction of 50% for each base material. Here as shown

in Figure 4.1, the initial material distribution consists eight soft-material elements at the

center of PBC while hard material assigned to all other elements.


84

Here in this section the objective is to optimize the topology of the microstructure so

that the materials bulk modulus K is maximized. In sample (a), Young’s moduli and

Poisson’s ratios are selected to be E1=1 and ν1=0.2 of soft base material; E2=10.0 and

ν2=0.45 of hard base material. Other design parameters include: evolutionary rate ER =

0.01, filter radius rmin = 2.5. Q interpolation function is chosen with penalty exponent p

= 3 for E, 1 for ν. Topology optimization starts with hard material as initial dominating

material, and carries out by gradually increasing volume of soft elements in PBC until

the volume constraint reaches 50% of each base material, and bulk modulus and

topology of composite approach convergence to their final solutions.

Figure 4.2 shows the evolution history of bulk modulus K and volume fraction Vf as a

function of iterations. As shown in Figure 4.2, bulk modulus decreases when elements

possessed by stronger base material also decrease. Once the volumetric constraint is

satisfied, the bulk modulus and micro-structural topology stably converge to the final

solutions.

Figure 4.1 Initial design.


85

The evolution of composite topology are presented in Figure 4.3 in the form of single

base cell, 2×2×2 base cell and their cross-section views respectively, at sampling

iterations 10, 25, 50 and 77. The optimization process took 77 iterations in total with

optimal bulk modulus of 4.2495. And the designed structure/material can be

constructed by repeating the presented microstructures.

Figure 4.2 Evolution history of bulk modulus and volume fraction for maximizing bulk modulus.


86

(a) iteration 10 (b) iteration 25 (c) iteration 50 (d) iteration 77

Figure 4.3 2×2×2 PBCs and cross-section views, single PBCs and cross-section views at (a) iteration 10, (b) iteration 25, (c) iteration 50, (d) iteration 77.

Sample (b) and (c) both contain base material of negative Poison’s ratio. The elasticity

properties of sample (b) are E1=1.0 and ν1= -0.9 of soft base material, E2=10.0 and

ν2=0.45 of hard base material. The elasticity properties of sample (c) are E1=1.0 and ν1=

-0.9 of soft base material, E2=10.0 and ν2= -0.5 of hard base material.


87

Table 4.1 Composites of maximum bulk moduli

Sample Base materials Bulk/shear moduli HSW upper bound (bulk

modulus)

Composite bulk

modulus Agreement

(a) E1=1.0 ν1=0.2

E2=10.0 ν2=0.45

K1=0.5556 G1=0.4167

K2=33.3333 G2=3.4483

4.4762 4.2495 95%

(b) E1=1.0 ν1= -0.9

E2=10.0 ν2=0.45

K1=0.1190 G1=5

K2=33.3333 G2=3.4483

4.9363 4.9071 99.41%

(c) E1=1.0 ν1= -0.9

E2=10.0 ν2= -0.5

K1=0.1190 G1=5

K2=1.6667 G2=10 0.8508 0.8379 98%

6.4241 3.1622 3.1622 0 0 0

3.1622 6.4241 3.1622 0 0 0 3.1622 3.1622 6.4241 0 0 0

0 0 0 1.3853 0 0 0 0 0 0 1.3853 0

0 0 0 0 0 1.3853

(a)

10.449 2.1361 2.1361 0 0 0

2.1361 10.449 2.1361 0 0 0 2.1361 2.1361 10.449 0 0 0

0 0 0 4.1359 0 0 0 0 0 0 4.1359 0

0 0 0 0 0 4.1359

(b)

10.257 -3.8713 -3.8713 0 0 0

-3.8713 10.257 -3.8713 0 0 0 -3.8713 -3.8713 10.257 0 0 0

0 0 0 6.9687 0 0 0 0 0 0 6.9687 0

0 0 0 0 0 6.9687

(c)

Figure 4.4 optimized topologies and effective elasticity matrices of three PBCs with maximum bulk modulus: case (a), (b) and (c).


88

Table 4.2 parameters and results for best solution under each condition (cubic symmetric model)

Maximising bulk

modulus

Base materials

Filter radius (rmin)

ER Mesh density

Interpolation function Iterations Bulk

modulus Shear

modulus

E1=1.0 ν1=0.2

E2=10.0 ν2=0.45

2.5 0.01 303 Q 77 4.2495 1.3853

E1=1.0 ν1= -0.9 E2=10.0 ν2=0.45

1.5 0.01 303 Q 66 4.9071 4.1359

E1=1.0 ν1= -0.9 E2=10.0 ν2=-0.5

3.0 0.01 303 SIMP 61 0.8379 6.9687

4.2.2 Composites with maximum shear modulus

Here in this section the objective is to optimize the topology of the microstructure so

that the materials shear modulus is maximized. Case studies use the same initial

design as in previous section as shown in Figure 4.1.

Below in sample (a), Young’s moduli and Poisson’s ratios are selected to be E1=1 and

ν1=0.2 of soft base material; E2=10.0 and ν2=0.45 of hard base material. Other design

parameters include: evolutionary rate ER = 0.01, filter radius rmin = 2.5. SIMP


89

interpolation function is chosen with penalty exponent p = 3 for E, 1 for ν. Topology

optimization starts with hard material as initial dominating material, and carries out by

gradually increasing volume of soft elements in PBC until the volume constraint

reaches 50% of each base material, and shear modulus and topology of composite

approach convergence to their final solutions.

Figure 4.5 shows the evolution history of shear modulus and volume fraction Vf as a

function of iterations. As shown in Figure 4.5, shear modulus decreases when elements

possessed by stronger base material also decrease. Once the volumetric constraint is

satisfied, the shear modulus and micro-structural topology stably converge to the final

solutions.

The evolution of composite topology are presented in Figure 4.6 in the form of single

base cell, 2×2×2 base cell and their cross-section views respectively, at sampling

iterations 8, 16, 24 and 33. The optimization process took 33 iterations in total with

optimal shear modulus of 1.5320. And the designed structure/material can be

constructed by repeating the presented microstructures.


90

Figure 4.5 Evolution history of shear modulus and volume fraction for maximizing shear modulus.


91

(a) iteration 8 (b) iteration 16 (c) iteration 24 (d) iteration 33

Figure 4.6 2×2×2 PBCs and cross-section views, single PBCs and cross-section views at (a) iteration 8, (b) iteration 16, (c) iteration 24, (d) iteration 33.

Sample (b) and (c) both contain base material of negative Poison’s ratio. The elasticity

properties of sample (b) are E1=1.0 and ν1= -0.9 of soft base material, E2=20.0 and

ν2=0.45 of hard base material. The elasticity properties of sample (c) are E1=1.0 and ν1=

-0.9 of soft base material, E2=10.0 and ν2= -0.5 of hard base material.


92

Table 4.3 Composites of maximum shear moduli

Sample Base materials Bulk/shear moduli HSW upper bound (shear modulus)

Composite shear modulus Agreement

(a) E1=1.0 ν1=0.2

E2=10.0 ν2=0.45

K1=0.5556 G1=0.4167

K2=33.3333 G2=3.4483 1.5850 1.5320 97%

(b) E1=1.0 ν1= -0.9 E2=20.0 ν2=0.45

K1=0.1190 G1=5 K2=66.6667 G2=6.8966 5.8895 5.7954 98%

(c) E1=1.0 ν1= -0.9

E2=10.0 ν2= -0.5

K1=0.1190 G1=5

K2=1.6667 G2=10 7.0779 6.9450 98%

5.616 2.9355 2.9355 0 0 0

2.9355 5.616 2.9355 0 0 0 2.9355 2.9355 5.616 0 0 0

0 0 0 1.532 0 0 0 0 0 0 1.532 0

0 0 0 0 0 1.532

(a)

13.546 1.9457 1.9457 0 0 0

1.9457 13.546 1.9457 0 0 0 1.9457 1.9457 13.546 0 0 0

0 0 0 5.7954 0 0 0 0 0 0 5.7954 0

0 0 0 0 0 5.7954

(b)

10.235 -3.8692 -3.8692 0 0 0

-3.8692 10.235 -3.8692 0 0 0 -3.8692 -3.8692 10.235 0 0 0

0 0 0 6.945 0 0 0 0 0 0 6.945 0

0 0 0 0 0 6.945

(c)

Figure 4.7 optimized topologies and effective elasticity matrices of three PBCs with maximum shear modulus: sample (a), (b) and (c).


93

Table 4.4 parameters and results for best solution under each condition (cubic symmetric model)

Maximising shear modulus

Base materials

Filter radius (rmin)

ER Mesh density

Interpolation function

Iterations

Bulk modulus

Shear modulu

s

E1=1.0 ν1=0.2 E2=10.0 ν2=0.45

3.0 0.02 303 SIMP 34 3.8290 1.5320

E1=1.0 ν1= -0.9 E2=20.0 ν2=0.45

3.0 0.02 303 SIMP 33 5.8123 5.7954

E1=1.0 ν1= -0.9 E2=10.0 ν2=-0.5

3.0 0.02 303 SIMP 68 0.8321 6.9450

4.2.3 Validation using HSW theoretical bound

The upper and lower bounds for bulk modulus for well-ordered quasi-homogenous and

quasi-isotropic composites have been derived by Hashin and Shtrikman, which is

refered to as Hashin Shtrikman (HS) bounds (Hashin and Shtrikman, 1963). As

mentioned at the beginning of this chapter, these bounds have been used for predicting

the range of properties that a composite can achieve based on given base materials and

volume fractions. They have also been applied broadly in the validation of topology

optimization of various microstructures (Challis, Roberts et al. 2008; Cadman, Zhou et

al. 2012).


94

Walpole (Walpole, L., 1966) developed another variational method that derived bounds

that did not require phases to be “well-ordered”. Nevertheless it applies to well-ordered

phases as well, and reaches same results as HS bounds in such cases. As indicated in

(Gibiansky and Sigmund 2000) the Hashin-Shtrikman-Walpole (HSW) bounds on the

bulk modulus are not only valid for isotropic materials but also applicable for materials

with square symmetry (in 2D cases) and cubic symmetry (in 3D cases). For composite

that are made of N elastic phases, with bulk modulus and shear modulus , and

prescribed volume fraction , The HSW analytical upper bound on bulk modulus and

shear modulus of composite materials can be expressed as (Gibiansky and Sigmund

2000):

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'()( *"+",-./$0 1( 2"345-% 6 7( 89:;( <;$.( 2=>"%. ?@ ABCCCD ?EFG?H@SAS

(4.30)

Here are the theoretical upper bound of bulk modulus and shear modulus,

are the maximal values of the base materials’ bulk and shear moduli,

respectively. d = 2 or d = 3 is the spatial dimension.

In order to examine optimal moduli according to Hashin-Shtrikman-Walpole (HSW)


95

bounds, here we choose two sets of base materials to form two-phase composites. The

Hashin-Shtrikman-Walpole (HSW) upper bounds on effective bulk and shear moduli of

potential composites can be calculated as shown in Table 4.5.

Table 4.5 Two composites and their corresponding HSW upper bounds

(Target volume=50%)

Sample Base materials Bulk/shear moduli HSW upper bound (bulk modulus)

HSW upper bound

(shear modulus)

(a) E1=1.0 ν1= 0.3

(E2=10-9 ν2=0.3)

K1= 0.8333 G1= 0.3846

K2=0 G2=0 0.2299 0.1322

(b) E1=1.0 ν1=0.2

E2=10.0 ν2=0.2

K1=0.5556 G1=0.4167

K2=5.5556 G2=4.1667 2.3297 1.7473

The HSW upper bounds of bulk modulus Ku are attainable subject to the restrictions:

If K1 ≤ K2 , the HSW upper bound is attainable if

G1 ≥G2, Ku ≤ K2 , (4.31a)

Or if

G2 ≥G1, Ku ≥ K1 , (4.31b)

and vice versa. Both composites here satisfy (4.31). Therefore their bulk moduli K have

the potential to achieve upper bounds Ku .


96

Similarly, the HSW upper bounds of shear moduli Gu are attainable under the

restrictions that: when G1 ≤G2 ,

K2 ≥ K1, Gu ≥G1 . (4.32)

and vice versa. So in both cases it’s possible to build composites with shear modulus

Gu .

Sample (a) is single-phase material. The mechanical properties of the solid phase are the

Young’s modulus, E = 1 and the Poisson’s ratio, 3.0=v . The BESO parameters are

selected as the evolutionary rate =ER 0.01, the filter radius rmin = 3 and the penalty

exponent 3=p . This single phase (solid - void) unit here is the simplest case of

composite material. Objective here is maximising bulk/shear modulus with volume

fraction 50% to examine the reliability/deviation of python code in this study. In Figure

4.8, bulk modulus of the composite is optimised.

0.3968 0.1309 0.1309 0 0 0

0.1309 0.3968 0.1309 0 0 0

0.1309 0.1309 0.3968 0 0 0

0 0 0 0.1056 0 0

0 0 0 0 0.1056 0

0 0 0 0 0 0.1056

Figure 4.8. Optimal design of sample (a) (SIMP interpolation) 0.2195.


97

The following case is two-phase composite material. Its mechanical properties are the

Young’s moduli, E1=1.0 E2=10.0 and the Poisson’s ratio, ν1=0.2 ν2=0.2 too. The BESO

parameters are chosen as the evolutionary rate =ER 0.01, the filter radius rmin = 3 and

the penalty exponent 3=p . In Figure 4.9, bulk modulus of the composite is optimised.

4.3515 0.8481 0.8481 0 0 0

0.8481 4.3515 0.8481 0 0 0

0.8481 0.8481 4.3515 0 0 0

0 0 0 1.3162 0 0

0 0 0 0 1.3162 0

0 0 0 0 0 1.3162

Figure 4.9. Optimal design of sample (b) (SIMP interpolation) 2.0159.

Below in Figure 4.10 and Figure 4.11, shear moduli of both samples are optimised.

0.3545 0.1511 0.1511 0 0 0

0.1511 0.3545 0.1511 0 0 0

0.1511 0.1511 0.3545 0 0 0

0 0 0 0.1305 0 0

0 0 0 0 0.1305 0

0 0 0 0 0 0.1305

Figure 4.10. Optimal design of sample (a) (SIMP interpolation) 0.1305.


98

4.2351 1.2419 1.2419 0 0 0

1.2419 4.2351 1.2419 0 0 0

1.2419 1.2419 4.2351 0 0 0

0 0 0 1.7349 0 0

0 0 0 0 1.7349 0

0 0 0 0 0 1.7349

Figure 4.11. Optimal design of sample (b) (SIMP interpolation) 1.7349.

Compare the attained effective optimal moduli with HSW upper bounds as seen in

Table 4.6 and Table 4.7, we can observe that they show very good agreement with the

analytical upper bounds.

Table 4.6 Composites of maximum bulk moduli

Case Base materials HSW upper bound (bulk

modulus)

Optimal bulk modulus (Python

method)

(a) E1=1.0 ν1= 0.3

(E2=10-9 ν2=0.3) 0.2299 0.2195

(b) E1=1.0 ν1=0.2

E2=10.0 ν2=0.2 2.3297 2.0159


99

Table 4.7 Composites of maximum shear moduli

Case Base materials HSW upper bound (shear

modulus)

Optimal shear modulus (Python

method)

(a) E1=1.0 ν1= 0.3

(E2=10-9 ν2=0.3) 0.1322 0.1305

(b) E1=1.0 ν1=0.2

E2=10.0 ν2=0.2 1.7473 1.7349

4.3. Concluding remarks

In this chapter the BESO method has been developed into the designing of two-phase

composites with maximum bulk modulus or shear modulus. The idea of BESO method

is to seek optimal material spatial distribution within periodic base cell by performing

topology optimization subject to volume constraints. With the help of homogenization

theory, the overall effective properties of design domain can be calculated. Sensitivity

analysis is carried out in each iteration also utilising the homogenization theory. The

binary design variable , which marks the density of elements, is redefined elementally

based on ranking of sensitivity numbers of all elements in PBC. The optimization

process gradually evolves towards on optimal solution as well as satisfying both

prescribed volume constraint and convergence criterion.

In this study ABAQUS was used as the FE analysis tool, and the BESO part is

implemented as a “post-processor” in Python code. Several case studies are presented to

demonstrate the effectiveness of the proposed BESO method. The known HSW


100

analytical bounds are used for verifications of the results. Some interesting topological

patterns have been found for guiding the composite material design. The method is the

first one that predicts optimal bulk/shear modulus of composites possessing base

materials of different Poisson’s ratio. The methodology developed in this chapter can be

further extended to various other material design scenarios easily.


101

Chapter5ConclusionsandRecommendations

5.1. Conclusions

This thesis has presented a new approach for the design of microstructures of materials

with different Poisson’s ratio using the BESO method. It is assumed that the materials

are composed of periodic base cells. A new algorithm for composite material design


102

with extreme Young's moduli has been developed and then the new procedure for

design of two-phase composite materials with base materials possessing different

Poisson's ratio has also been proposed. The main conclusions can be obtained from this

thesis as follows,

�� a computational algorithm for topological design of composite materials with

extreme Young’s moduli has been developed. The composite consists of base materials

with different Poisson’s ratios. The elemental sensitivity analysis has been carried out

using finite element modelling and the homogenization theory. Based on the sensitivity

ranking, the optimal topology for the PBC is obtained. Numerical results show that the

proposed algorithm is effective and computational efficient. The modulus of the

composite is much higher than that of base materials. When one of the base materials

has negative Poisson's ratio, the modulus of the composite increases significantly.

�� a computational algorithm for topological design of two-phase composite

materials with extreme bulk or shear modulus has been developed. These composites

possess two different materials phases. The algorithm is to achieve the optimal material

spatial distribution within periodic base cell by performing topology optimization

subject to the volume constraints. The homogenization theory is adopted to build the

relationship between the material properties of the whole design body in macrostructure

scope and that of the periodic base cell in microstructure scope. The sensitivity analysis

is carried out in each iteration to achieve an optimal solution. Several case studies have

been presented to demonstrate the effectiveness of the proposed algorithm.


103

5.2. Recommendations

The approach for the design of microstructures of materials with different Poisson’s

ratio using the BESO method has been presented in this thesis. Several case studies

have been carried out to show the effectiveness and computation efficiency of the

proposed approach. Some further work is listed as follows:

�� More case studies should be carried out next step to broaden applications of the

developed algorithm.

�� Other objective functions such as frequency and thermal expansion could be

considered.

�� It would be worthwhile extending the present study to the concurrent

optimisation of both macrostructure and material microstructure of composites with

multiple phases of distinct Poisson’s ratios.


104

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